Fixed database typo and removed unnecessary class identifier.

venv/Lib/site-packages/scipy/spatial/__init__.py

@@ -0,0 +1,111 @@
+"""
+=============================================================
+Spatial algorithms and data structures (:mod:`scipy.spatial`)
+=============================================================
+
+.. currentmodule:: scipy.spatial
+
+Spatial transformations
+=======================
+
+These are contained in the `scipy.spatial.transform` submodule.
+
+Nearest-neighbor queries
+========================
+.. autosummary::
+   :toctree: generated/
+
+   KDTree      -- class for efficient nearest-neighbor queries
+   cKDTree     -- class for efficient nearest-neighbor queries (faster implementation)
+   Rectangle
+
+Distance metrics are contained in the :mod:`scipy.spatial.distance` submodule.
+
+Delaunay triangulation, convex hulls, and Voronoi diagrams
+==========================================================
+
+.. autosummary::
+   :toctree: generated/
+
+   Delaunay    -- compute Delaunay triangulation of input points
+   ConvexHull  -- compute a convex hull for input points
+   Voronoi     -- compute a Voronoi diagram hull from input points
+   SphericalVoronoi -- compute a Voronoi diagram from input points on the surface of a sphere
+   HalfspaceIntersection -- compute the intersection points of input halfspaces
+
+Plotting helpers
+================
+
+.. autosummary::
+   :toctree: generated/
+
+   delaunay_plot_2d     -- plot 2-D triangulation
+   convex_hull_plot_2d  -- plot 2-D convex hull
+   voronoi_plot_2d      -- plot 2-D Voronoi diagram
+
+.. seealso:: :ref:`Tutorial <qhulltutorial>`
+
+
+Simplex representation
+======================
+The simplices (triangles, tetrahedra, etc.) appearing in the Delaunay
+tessellation (N-D simplices), convex hull facets, and Voronoi ridges
+(N-1-D simplices) are represented in the following scheme::
+
+    tess = Delaunay(points)
+    hull = ConvexHull(points)
+    voro = Voronoi(points)
+
+    # coordinates of the jth vertex of the ith simplex
+    tess.points[tess.simplices[i, j], :]        # tessellation element
+    hull.points[hull.simplices[i, j], :]        # convex hull facet
+    voro.vertices[voro.ridge_vertices[i, j], :] # ridge between Voronoi cells
+
+For Delaunay triangulations and convex hulls, the neighborhood
+structure of the simplices satisfies the condition:
+``tess.neighbors[i,j]`` is the neighboring simplex of the ith
+simplex, opposite to the ``j``-vertex. It is -1 in case of no neighbor.
+
+Convex hull facets also define a hyperplane equation::
+
+    (hull.equations[i,:-1] * coord).sum() + hull.equations[i,-1] == 0
+
+Similar hyperplane equations for the Delaunay triangulation correspond
+to the convex hull facets on the corresponding N+1-D
+paraboloid.
+
+The Delaunay triangulation objects offer a method for locating the
+simplex containing a given point, and barycentric coordinate
+computations.
+
+Functions
+---------
+
+.. autosummary::
+   :toctree: generated/
+
+   tsearch
+   distance_matrix
+   minkowski_distance
+   minkowski_distance_p
+   procrustes
+   geometric_slerp
+
+"""
+
+from .kdtree import *
+from .ckdtree import *
+from .qhull import *
+from ._spherical_voronoi import SphericalVoronoi
+from ._plotutils import *
+from ._procrustes import procrustes
+from ._geometric_slerp import geometric_slerp
+
+__all__ = [s for s in dir() if not s.startswith('_')]
+__all__ += ['distance', 'transform']
+
+from . import distance, transform
+
+from scipy._lib._testutils import PytestTester
+test = PytestTester(__name__)
+del PytestTester

venv/Lib/site-packages/scipy/spatial/_geometric_slerp.py

@@ -0,0 +1,227 @@
+from __future__ import division, print_function, absolute_import
+
+__all__ = ['geometric_slerp']
+
+import warnings
+
+import numpy as np
+from scipy.spatial.distance import euclidean
+
+
+def _geometric_slerp(start, end, t):
+    # create an orthogonal basis using QR decomposition
+    basis = np.vstack([start, end])
+    Q, R = np.linalg.qr(basis.T)
+    signs = 2 * (np.diag(R) >= 0) - 1
+    Q = Q.T * signs.T[:, np.newaxis]
+    R = R.T * signs.T[:, np.newaxis]
+
+    # calculate the angle between `start` and `end`
+    c = np.dot(start, end)
+    s = np.linalg.det(R)
+    omega = np.arctan2(s, c)
+
+    # interpolate
+    start, end = Q
+    s = np.sin(t * omega)
+    c = np.cos(t * omega)
+    return start * c[:, np.newaxis] + end * s[:, np.newaxis]
+
+
+def geometric_slerp(start,
+                    end,
+                    t,
+                    tol=1e-7):
+    """
+    Geometric spherical linear interpolation.
+
+    The interpolation occurs along a unit-radius
+    great circle arc in arbitrary dimensional space.
+
+    Parameters
+    ----------
+    start : (n_dimensions, ) array-like
+        Single n-dimensional input coordinate in a 1-D array-like
+        object. `n` must be greater than 1.
+    end : (n_dimensions, ) array-like
+        Single n-dimensional input coordinate in a 1-D array-like
+        object. `n` must be greater than 1.
+    t: float or (n_points,) array-like
+        A float or array-like of doubles representing interpolation
+        parameters, with values required in the inclusive interval
+        between 0 and 1. A common approach is to generate the array
+        with ``np.linspace(0, 1, n_pts)`` for linearly spaced points.
+        Ascending, descending, and scrambled orders are permitted.
+    tol: float
+        The absolute tolerance for determining if the start and end
+        coordinates are antipodes.
+
+    Returns
+    -------
+    result : (t.size, D)
+        An array of doubles containing the interpolated
+        spherical path and including start and
+        end when 0 and 1 t are used. The
+        interpolated values should correspond to the
+        same sort order provided in the t array. The result
+        may be 1-dimensional if ``t`` is a float.
+
+    Raises
+    ------
+    ValueError
+        If ``start`` and ``end`` are antipodes, not on the
+        unit n-sphere, or for a variety of degenerate conditions.
+
+    Notes
+    -----
+    The implementation is based on the mathematical formula provided in [1]_,
+    and the first known presentation of this algorithm, derived from study of
+    4-D geometry, is credited to Glenn Davis in a footnote of the original
+    quaternion Slerp publication by Ken Shoemake [2]_.
+
+    .. versionadded:: 1.5.0
+
+    References
+    ----------
+    .. [1] https://en.wikipedia.org/wiki/Slerp#Geometric_Slerp
+    .. [2] Ken Shoemake (1985) Animating rotation with quaternion curves.
+           ACM SIGGRAPH Computer Graphics, 19(3): 245-254.
+
+    See Also
+    --------
+    scipy.spatial.transform.Slerp : 3-D Slerp that works with quaternions
+
+    Examples
+    --------
+    Interpolate four linearly-spaced values on the circumference of
+    a circle spanning 90 degrees:
+
+    >>> from scipy.spatial import geometric_slerp
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> ax = fig.add_subplot(111)
+    >>> start = np.array([1, 0])
+    >>> end = np.array([0, 1])
+    >>> t_vals = np.linspace(0, 1, 4)
+    >>> result = geometric_slerp(start,
+    ...                          end,
+    ...                          t_vals)
+
+    The interpolated results should be at 30 degree intervals
+    recognizable on the unit circle:
+
+    >>> ax.scatter(result[...,0], result[...,1], c='k')
+    >>> circle = plt.Circle((0, 0), 1, color='grey')
+    >>> ax.add_artist(circle)
+    >>> ax.set_aspect('equal')
+    >>> plt.show()
+
+    Attempting to interpolate between antipodes on a circle is
+    ambiguous because there are two possible paths, and on a
+    sphere there are infinite possible paths on the geodesic surface.
+    Nonetheless, one of the ambiguous paths is returned along
+    with a warning:
+
+    >>> opposite_pole = np.array([-1, 0])
+    >>> with np.testing.suppress_warnings() as sup:
+    ...     sup.filter(UserWarning)
+    ...     geometric_slerp(start,
+    ...                     opposite_pole,
+    ...                     t_vals)
+    array([[ 1.00000000e+00,  0.00000000e+00],
+           [ 5.00000000e-01,  8.66025404e-01],
+           [-5.00000000e-01,  8.66025404e-01],
+           [-1.00000000e+00,  1.22464680e-16]])
+
+    Extend the original example to a sphere and plot interpolation
+    points in 3D:
+
+    >>> from mpl_toolkits.mplot3d import proj3d
+    >>> fig = plt.figure()
+    >>> ax = fig.add_subplot(111, projection='3d')
+
+    Plot the unit sphere for reference (optional):
+
+    >>> u = np.linspace(0, 2 * np.pi, 100)
+    >>> v = np.linspace(0, np.pi, 100)
+    >>> x = np.outer(np.cos(u), np.sin(v))
+    >>> y = np.outer(np.sin(u), np.sin(v))
+    >>> z = np.outer(np.ones(np.size(u)), np.cos(v))
+    >>> ax.plot_surface(x, y, z, color='y', alpha=0.1)
+
+    Interpolating over a larger number of points
+    may provide the appearance of a smooth curve on
+    the surface of the sphere, which is also useful
+    for discretized integration calculations on a
+    sphere surface:
+
+    >>> start = np.array([1, 0, 0])
+    >>> end = np.array([0, 0, 1])
+    >>> t_vals = np.linspace(0, 1, 200)
+    >>> result = geometric_slerp(start,
+    ...                          end,
+    ...                          t_vals)
+    >>> ax.plot(result[...,0],
+    ...         result[...,1],
+    ...         result[...,2],
+    ...         c='k')
+    >>> plt.show()
+    """
+
+    start = np.asarray(start, dtype=np.float64)
+    end = np.asarray(end, dtype=np.float64)
+
+    if start.ndim != 1 or end.ndim != 1:
+        raise ValueError("Start and end coordinates "
+                         "must be one-dimensional")
+
+    if start.size != end.size:
+        raise ValueError("The dimensions of start and "
+                         "end must match (have same size)")
+
+    if start.size < 2 or end.size < 2:
+        raise ValueError("The start and end coordinates must "
+                         "both be in at least two-dimensional "
+                         "space")
+
+    if np.array_equal(start, end):
+        return [start] * np.asarray(t).size
+
+    # for points that violate equation for n-sphere
+    for coord in [start, end]:
+        if not np.allclose(np.linalg.norm(coord), 1.0,
+                           rtol=1e-9,
+                           atol=0):
+            raise ValueError("start and end are not"
+                             " on a unit n-sphere")
+
+    if not isinstance(tol, float):
+        raise ValueError("tol must be a float")
+    else:
+        tol = np.fabs(tol)
+
+    coord_dist = euclidean(start, end)
+
+    # diameter of 2 within tolerance means antipodes, which is a problem
+    # for all unit n-spheres (even the 0-sphere would have an ambiguous path)
+    if np.allclose(coord_dist, 2.0, rtol=0, atol=tol):
+        warnings.warn("start and end are antipodes"
+                      " using the specified tolerance;"
+                      " this may cause ambiguous slerp paths")
+
+    t = np.asarray(t, dtype=np.float64)
+
+    if t.size == 0:
+        return np.empty((0, start.size))
+
+    if t.min() < 0 or t.max() > 1:
+        raise ValueError("interpolation parameter must be in [0, 1]")
+
+    if t.ndim == 0:
+        return _geometric_slerp(start,
+                                end,
+                                np.atleast_1d(t)).ravel()
+    else:
+        return _geometric_slerp(start,
+                                end,
+                                t)

venv/Lib/site-packages/scipy/spatial/_plotutils.py

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+import numpy as np
+from scipy._lib.decorator import decorator as _decorator
+
+__all__ = ['delaunay_plot_2d', 'convex_hull_plot_2d', 'voronoi_plot_2d']
+
+
+@_decorator
+def _held_figure(func, obj, ax=None, **kw):
+    import matplotlib.pyplot as plt  # type: ignore[import]
+
+    if ax is None:
+        fig = plt.figure()
+        ax = fig.gca()
+        return func(obj, ax=ax, **kw)
+
+    # As of matplotlib 2.0, the "hold" mechanism is deprecated.
+    # When matplotlib 1.x is no longer supported, this check can be removed.
+    was_held = getattr(ax, 'ishold', lambda: True)()
+    if was_held:
+        return func(obj, ax=ax, **kw)
+    try:
+        ax.hold(True)
+        return func(obj, ax=ax, **kw)
+    finally:
+        ax.hold(was_held)
+
+
+def _adjust_bounds(ax, points):
+    margin = 0.1 * points.ptp(axis=0)
+    xy_min = points.min(axis=0) - margin
+    xy_max = points.max(axis=0) + margin
+    ax.set_xlim(xy_min[0], xy_max[0])
+    ax.set_ylim(xy_min[1], xy_max[1])
+
+
+@_held_figure
+def delaunay_plot_2d(tri, ax=None):
+    """
+    Plot the given Delaunay triangulation in 2-D
+
+    Parameters
+    ----------
+    tri : scipy.spatial.Delaunay instance
+        Triangulation to plot
+    ax : matplotlib.axes.Axes instance, optional
+        Axes to plot on
+
+    Returns
+    -------
+    fig : matplotlib.figure.Figure instance
+        Figure for the plot
+
+    See Also
+    --------
+    Delaunay
+    matplotlib.pyplot.triplot
+
+    Notes
+    -----
+    Requires Matplotlib.
+
+    Examples
+    --------
+
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.spatial import Delaunay, delaunay_plot_2d
+
+    The Delaunay triangulation of a set of random points:
+
+    >>> points = np.random.rand(30, 2)
+    >>> tri = Delaunay(points)
+
+    Plot it:
+
+    >>> _ = delaunay_plot_2d(tri)
+    >>> plt.show()
+
+    """
+    if tri.points.shape[1] != 2:
+        raise ValueError("Delaunay triangulation is not 2-D")
+
+    x, y = tri.points.T
+    ax.plot(x, y, 'o')
+    ax.triplot(x, y, tri.simplices.copy())
+
+    _adjust_bounds(ax, tri.points)
+
+    return ax.figure
+
+
+@_held_figure
+def convex_hull_plot_2d(hull, ax=None):
+    """
+    Plot the given convex hull diagram in 2-D
+
+    Parameters
+    ----------
+    hull : scipy.spatial.ConvexHull instance
+        Convex hull to plot
+    ax : matplotlib.axes.Axes instance, optional
+        Axes to plot on
+
+    Returns
+    -------
+    fig : matplotlib.figure.Figure instance
+        Figure for the plot
+
+    See Also
+    --------
+    ConvexHull
+
+    Notes
+    -----
+    Requires Matplotlib.
+
+
+    Examples
+    --------
+
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.spatial import ConvexHull, convex_hull_plot_2d
+
+    The convex hull of a random set of points:
+
+    >>> points = np.random.rand(30, 2)
+    >>> hull = ConvexHull(points)
+
+    Plot it:
+
+    >>> _ = convex_hull_plot_2d(hull)
+    >>> plt.show()
+
+    """
+    from matplotlib.collections import LineCollection  # type: ignore[import]
+
+    if hull.points.shape[1] != 2:
+        raise ValueError("Convex hull is not 2-D")
+
+    ax.plot(hull.points[:,0], hull.points[:,1], 'o')
+    line_segments = [hull.points[simplex] for simplex in hull.simplices]
+    ax.add_collection(LineCollection(line_segments,
+                                     colors='k',
+                                     linestyle='solid'))
+    _adjust_bounds(ax, hull.points)
+
+    return ax.figure
+
+
+@_held_figure
+def voronoi_plot_2d(vor, ax=None, **kw):
+    """
+    Plot the given Voronoi diagram in 2-D
+
+    Parameters
+    ----------
+    vor : scipy.spatial.Voronoi instance
+        Diagram to plot
+    ax : matplotlib.axes.Axes instance, optional
+        Axes to plot on
+    show_points: bool, optional
+        Add the Voronoi points to the plot.
+    show_vertices : bool, optional
+        Add the Voronoi vertices to the plot.
+    line_colors : string, optional
+        Specifies the line color for polygon boundaries
+    line_width : float, optional
+        Specifies the line width for polygon boundaries
+    line_alpha: float, optional
+        Specifies the line alpha for polygon boundaries
+    point_size: float, optional
+        Specifies the size of points
+
+
+    Returns
+    -------
+    fig : matplotlib.figure.Figure instance
+        Figure for the plot
+
+    See Also
+    --------
+    Voronoi
+
+    Notes
+    -----
+    Requires Matplotlib.
+
+    Examples
+    --------
+    Set of point:
+
+    >>> import matplotlib.pyplot as plt
+    >>> points = np.random.rand(10,2) #random
+
+    Voronoi diagram of the points:
+
+    >>> from scipy.spatial import Voronoi, voronoi_plot_2d
+    >>> vor = Voronoi(points)
+
+    using `voronoi_plot_2d` for visualisation:
+
+    >>> fig = voronoi_plot_2d(vor)
+
+    using `voronoi_plot_2d` for visualisation with enhancements:
+
+    >>> fig = voronoi_plot_2d(vor, show_vertices=False, line_colors='orange',
+    ...                 line_width=2, line_alpha=0.6, point_size=2)
+    >>> plt.show()
+
+    """
+    from matplotlib.collections import LineCollection
+
+    if vor.points.shape[1] != 2:
+        raise ValueError("Voronoi diagram is not 2-D")
+
+    if kw.get('show_points', True):
+        point_size = kw.get('point_size', None)
+        ax.plot(vor.points[:,0], vor.points[:,1], '.', markersize=point_size)
+    if kw.get('show_vertices', True):
+        ax.plot(vor.vertices[:,0], vor.vertices[:,1], 'o')
+
+    line_colors = kw.get('line_colors', 'k')
+    line_width = kw.get('line_width', 1.0)
+    line_alpha = kw.get('line_alpha', 1.0)
+
+    center = vor.points.mean(axis=0)
+    ptp_bound = vor.points.ptp(axis=0)
+
+    finite_segments = []
+    infinite_segments = []
+    for pointidx, simplex in zip(vor.ridge_points, vor.ridge_vertices):
+        simplex = np.asarray(simplex)
+        if np.all(simplex >= 0):
+            finite_segments.append(vor.vertices[simplex])
+        else:
+            i = simplex[simplex >= 0][0]  # finite end Voronoi vertex
+
+            t = vor.points[pointidx[1]] - vor.points[pointidx[0]]  # tangent
+            t /= np.linalg.norm(t)
+            n = np.array([-t[1], t[0]])  # normal
+
+            midpoint = vor.points[pointidx].mean(axis=0)
+            direction = np.sign(np.dot(midpoint - center, n)) * n
+            if (vor.furthest_site):
+                direction = -direction
+            far_point = vor.vertices[i] + direction * ptp_bound.max()
+
+            infinite_segments.append([vor.vertices[i], far_point])
+
+    ax.add_collection(LineCollection(finite_segments,
+                                     colors=line_colors,
+                                     lw=line_width,
+                                     alpha=line_alpha,
+                                     linestyle='solid'))
+    ax.add_collection(LineCollection(infinite_segments,
+                                     colors=line_colors,
+                                     lw=line_width,
+                                     alpha=line_alpha,
+                                     linestyle='dashed'))
+
+    _adjust_bounds(ax, vor.points)
+
+    return ax.figure

venv/Lib/site-packages/scipy/spatial/_procrustes.py

@@ -0,0 +1,131 @@
+"""
+This module provides functions to perform full Procrustes analysis.
+
+This code was originally written by Justin Kucynski and ported over from
+scikit-bio by Yoshiki Vazquez-Baeza.
+"""
+
+import numpy as np
+from scipy.linalg import orthogonal_procrustes
+
+
+__all__ = ['procrustes']
+
+
+def procrustes(data1, data2):
+    r"""Procrustes analysis, a similarity test for two data sets.
+
+    Each input matrix is a set of points or vectors (the rows of the matrix).
+    The dimension of the space is the number of columns of each matrix. Given
+    two identically sized matrices, procrustes standardizes both such that:
+
+    - :math:`tr(AA^{T}) = 1`.
+
+    - Both sets of points are centered around the origin.
+
+    Procrustes ([1]_, [2]_) then applies the optimal transform to the second
+    matrix (including scaling/dilation, rotations, and reflections) to minimize
+    :math:`M^{2}=\sum(data1-data2)^{2}`, or the sum of the squares of the
+    pointwise differences between the two input datasets.
+
+    This function was not designed to handle datasets with different numbers of
+    datapoints (rows).  If two data sets have different dimensionality
+    (different number of columns), simply add columns of zeros to the smaller
+    of the two.
+
+    Parameters
+    ----------
+    data1 : array_like
+        Matrix, n rows represent points in k (columns) space `data1` is the
+        reference data, after it is standardised, the data from `data2` will be
+        transformed to fit the pattern in `data1` (must have >1 unique points).
+    data2 : array_like
+        n rows of data in k space to be fit to `data1`.  Must be the  same
+        shape ``(numrows, numcols)`` as data1 (must have >1 unique points).
+
+    Returns
+    -------
+    mtx1 : array_like
+        A standardized version of `data1`.
+    mtx2 : array_like
+        The orientation of `data2` that best fits `data1`. Centered, but not
+        necessarily :math:`tr(AA^{T}) = 1`.
+    disparity : float
+        :math:`M^{2}` as defined above.
+
+    Raises
+    ------
+    ValueError
+        If the input arrays are not two-dimensional.
+        If the shape of the input arrays is different.
+        If the input arrays have zero columns or zero rows.
+
+    See Also
+    --------
+    scipy.linalg.orthogonal_procrustes
+    scipy.spatial.distance.directed_hausdorff : Another similarity test
+      for two data sets
+
+    Notes
+    -----
+    - The disparity should not depend on the order of the input matrices, but
+      the output matrices will, as only the first output matrix is guaranteed
+      to be scaled such that :math:`tr(AA^{T}) = 1`.
+
+    - Duplicate data points are generally ok, duplicating a data point will
+      increase its effect on the procrustes fit.
+
+    - The disparity scales as the number of points per input matrix.
+
+    References
+    ----------
+    .. [1] Krzanowski, W. J. (2000). "Principles of Multivariate analysis".
+    .. [2] Gower, J. C. (1975). "Generalized procrustes analysis".
+
+    Examples
+    --------
+    >>> from scipy.spatial import procrustes
+
+    The matrix ``b`` is a rotated, shifted, scaled and mirrored version of
+    ``a`` here:
+
+    >>> a = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], 'd')
+    >>> b = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], 'd')
+    >>> mtx1, mtx2, disparity = procrustes(a, b)
+    >>> round(disparity)
+    0.0
+
+    """
+    mtx1 = np.array(data1, dtype=np.double, copy=True)
+    mtx2 = np.array(data2, dtype=np.double, copy=True)
+
+    if mtx1.ndim != 2 or mtx2.ndim != 2:
+        raise ValueError("Input matrices must be two-dimensional")
+    if mtx1.shape != mtx2.shape:
+        raise ValueError("Input matrices must be of same shape")
+    if mtx1.size == 0:
+        raise ValueError("Input matrices must be >0 rows and >0 cols")
+
+    # translate all the data to the origin
+    mtx1 -= np.mean(mtx1, 0)
+    mtx2 -= np.mean(mtx2, 0)
+
+    norm1 = np.linalg.norm(mtx1)
+    norm2 = np.linalg.norm(mtx2)
+
+    if norm1 == 0 or norm2 == 0:
+        raise ValueError("Input matrices must contain >1 unique points")
+
+    # change scaling of data (in rows) such that trace(mtx*mtx') = 1
+    mtx1 /= norm1
+    mtx2 /= norm2
+
+    # transform mtx2 to minimize disparity
+    R, s = orthogonal_procrustes(mtx1, mtx2)
+    mtx2 = np.dot(mtx2, R.T) * s
+
+    # measure the dissimilarity between the two datasets
+    disparity = np.sum(np.square(mtx1 - mtx2))
+
+    return mtx1, mtx2, disparity
+

venv/Lib/site-packages/scipy/spatial/_spherical_voronoi.py

@@ -0,0 +1,345 @@
+"""
+Spherical Voronoi Code
+
+.. versionadded:: 0.18.0
+
+"""
+#
+# Copyright (C)  Tyler Reddy, Ross Hemsley, Edd Edmondson,
+#                Nikolai Nowaczyk, Joe Pitt-Francis, 2015.
+#
+# Distributed under the same BSD license as SciPy.
+#
+
+import warnings
+import numpy as np
+import scipy
+from . import _voronoi
+from scipy.spatial import cKDTree
+
+__all__ = ['SphericalVoronoi']
+
+
+def calculate_solid_angles(R):
+    """Calculates the solid angles of plane triangles. Implements the method of
+    Van Oosterom and Strackee [VanOosterom]_ with some modifications. Assumes
+    that input points have unit norm."""
+    # Original method uses a triple product `R1 . (R2 x R3)` for the numerator.
+    # This is equal to the determinant of the matrix [R1 R2 R3], which can be
+    # computed with better stability.
+    numerator = np.linalg.det(R)
+    denominator = 1 + (np.einsum('ij,ij->i', R[:, 0], R[:, 1]) +
+                       np.einsum('ij,ij->i', R[:, 1], R[:, 2]) +
+                       np.einsum('ij,ij->i', R[:, 2], R[:, 0]))
+    return np.abs(2 * np.arctan2(numerator, denominator))
+
+
+class SphericalVoronoi:
+    """ Voronoi diagrams on the surface of a sphere.
+
+    .. versionadded:: 0.18.0
+
+    Parameters
+    ----------
+    points : ndarray of floats, shape (npoints, ndim)
+        Coordinates of points from which to construct a spherical
+        Voronoi diagram.
+    radius : float, optional
+        Radius of the sphere (Default: 1)
+    center : ndarray of floats, shape (ndim,)
+        Center of sphere (Default: origin)
+    threshold : float
+        Threshold for detecting duplicate points and
+        mismatches between points and sphere parameters.
+        (Default: 1e-06)
+
+    Attributes
+    ----------
+    points : double array of shape (npoints, ndim)
+        the points in `ndim` dimensions to generate the Voronoi diagram from
+    radius : double
+        radius of the sphere
+    center : double array of shape (ndim,)
+        center of the sphere
+    vertices : double array of shape (nvertices, ndim)
+        Voronoi vertices corresponding to points
+    regions : list of list of integers of shape (npoints, _ )
+        the n-th entry is a list consisting of the indices
+        of the vertices belonging to the n-th point in points
+
+    Methods
+    ----------
+    calculate_areas
+        Calculates the areas of the Voronoi regions. For 2D point sets, the
+        regions are circular arcs. The sum of the areas is `2 * pi * radius`.
+        For 3D point sets, the regions are spherical polygons. The sum of the
+        areas is `4 * pi * radius**2`.
+
+    Raises
+    ------
+    ValueError
+        If there are duplicates in `points`.
+        If the provided `radius` is not consistent with `points`.
+
+    Notes
+    -----
+    The spherical Voronoi diagram algorithm proceeds as follows. The Convex
+    Hull of the input points (generators) is calculated, and is equivalent to
+    their Delaunay triangulation on the surface of the sphere [Caroli]_.
+    The Convex Hull neighbour information is then used to
+    order the Voronoi region vertices around each generator. The latter
+    approach is substantially less sensitive to floating point issues than
+    angle-based methods of Voronoi region vertex sorting.
+
+    Empirical assessment of spherical Voronoi algorithm performance suggests
+    quadratic time complexity (loglinear is optimal, but algorithms are more
+    challenging to implement).
+
+    References
+    ----------
+    .. [Caroli] Caroli et al. Robust and Efficient Delaunay triangulations of
+                points on or close to a sphere. Research Report RR-7004, 2009.
+
+    .. [VanOosterom] Van Oosterom and Strackee. The solid angle of a plane
+                     triangle. IEEE Transactions on Biomedical Engineering,
+                     2, 1983, pp 125--126.
+
+    See Also
+    --------
+    Voronoi : Conventional Voronoi diagrams in N dimensions.
+
+    Examples
+    --------
+    Do some imports and take some points on a cube:
+
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy.spatial import SphericalVoronoi, geometric_slerp
+    >>> from mpl_toolkits.mplot3d import proj3d
+    >>> # set input data
+    >>> points = np.array([[0, 0, 1], [0, 0, -1], [1, 0, 0],
+    ...                    [0, 1, 0], [0, -1, 0], [-1, 0, 0], ])
+
+    Calculate the spherical Voronoi diagram:
+
+    >>> radius = 1
+    >>> center = np.array([0, 0, 0])
+    >>> sv = SphericalVoronoi(points, radius, center)
+
+    Generate plot:
+
+    >>> # sort vertices (optional, helpful for plotting)
+    >>> sv.sort_vertices_of_regions()
+    >>> t_vals = np.linspace(0, 1, 2000)
+    >>> fig = plt.figure()
+    >>> ax = fig.add_subplot(111, projection='3d')
+    >>> # plot the unit sphere for reference (optional)
+    >>> u = np.linspace(0, 2 * np.pi, 100)
+    >>> v = np.linspace(0, np.pi, 100)
+    >>> x = np.outer(np.cos(u), np.sin(v))
+    >>> y = np.outer(np.sin(u), np.sin(v))
+    >>> z = np.outer(np.ones(np.size(u)), np.cos(v))
+    >>> ax.plot_surface(x, y, z, color='y', alpha=0.1)
+    >>> # plot generator points
+    >>> ax.scatter(points[:, 0], points[:, 1], points[:, 2], c='b')
+    >>> # plot Voronoi vertices
+    >>> ax.scatter(sv.vertices[:, 0], sv.vertices[:, 1], sv.vertices[:, 2],
+    ...                    c='g')
+    >>> # indicate Voronoi regions (as Euclidean polygons)
+    >>> for region in sv.regions:
+    ...    n = len(region)
+    ...    for i in range(n):
+    ...        start = sv.vertices[region][i]
+    ...        end = sv.vertices[region][(i + 1) % n]
+    ...        result = geometric_slerp(start, end, t_vals)
+    ...        ax.plot(result[..., 0],
+    ...                result[..., 1],
+    ...                result[..., 2],
+    ...                c='k')
+    >>> ax.azim = 10
+    >>> ax.elev = 40
+    >>> _ = ax.set_xticks([])
+    >>> _ = ax.set_yticks([])
+    >>> _ = ax.set_zticks([])
+    >>> fig.set_size_inches(4, 4)
+    >>> plt.show()
+
+    """
+    def __init__(self, points, radius=1, center=None, threshold=1e-06):
+
+        if radius is None:
+            radius = 1.
+            warnings.warn('`radius` is `None`. '
+                          'This will raise an error in a future version. '
+                          'Please provide a floating point number '
+                          '(i.e. `radius=1`).',
+                          DeprecationWarning)
+
+        self.radius = float(radius)
+        self.points = np.array(points).astype(np.double)
+        self._dim = len(points[0])
+        if center is None:
+            self.center = np.zeros(self._dim)
+        else:
+            self.center = np.array(center, dtype=float)
+
+        # test degenerate input
+        self._rank = np.linalg.matrix_rank(self.points - self.points[0],
+                                           tol=threshold * self.radius)
+        if self._rank < self._dim:
+            raise ValueError("Rank of input points must be at least {0}".format(self._dim))
+
+        if cKDTree(self.points).query_pairs(threshold * self.radius):
+            raise ValueError("Duplicate generators present.")
+
+        radii = np.linalg.norm(self.points - self.center, axis=1)
+        max_discrepancy = np.abs(radii - self.radius).max()
+        if max_discrepancy >= threshold * self.radius:
+            raise ValueError("Radius inconsistent with generators.")
+
+        self._calc_vertices_regions()
+
+    def _calc_vertices_regions(self):
+        """
+        Calculates the Voronoi vertices and regions of the generators stored
+        in self.points. The vertices will be stored in self.vertices and the
+        regions in self.regions.
+
+        This algorithm was discussed at PyData London 2015 by
+        Tyler Reddy, Ross Hemsley and Nikolai Nowaczyk
+        """
+        # get Convex Hull
+        conv = scipy.spatial.ConvexHull(self.points)
+        # get circumcenters of Convex Hull triangles from facet equations
+        # for 3D input circumcenters will have shape: (2N-4, 3)
+        self.vertices = self.radius * conv.equations[:, :-1] + self.center
+        self._simplices = conv.simplices
+        # calculate regions from triangulation
+        # for 3D input simplex_indices will have shape: (2N-4,)
+        simplex_indices = np.arange(len(self._simplices))
+        # for 3D input tri_indices will have shape: (6N-12,)
+        tri_indices = np.column_stack([simplex_indices] * self._dim).ravel()
+        # for 3D input point_indices will have shape: (6N-12,)
+        point_indices = self._simplices.ravel()
+        # for 3D input indices will have shape: (6N-12,)
+        indices = np.argsort(point_indices, kind='mergesort')
+        # for 3D input flattened_groups will have shape: (6N-12,)
+        flattened_groups = tri_indices[indices].astype(np.intp)
+        # intervals will have shape: (N+1,)
+        intervals = np.cumsum(np.bincount(point_indices + 1))
+        # split flattened groups to get nested list of unsorted regions
+        groups = [list(flattened_groups[intervals[i]:intervals[i + 1]])
+                  for i in range(len(intervals) - 1)]
+        self.regions = groups
+
+    def sort_vertices_of_regions(self):
+        """Sort indices of the vertices to be (counter-)clockwise ordered.
+
+        Raises
+        ------
+        TypeError
+            If the points are not three-dimensional.
+
+        Notes
+        -----
+        For each region in regions, it sorts the indices of the Voronoi
+        vertices such that the resulting points are in a clockwise or
+        counterclockwise order around the generator point.
+
+        This is done as follows: Recall that the n-th region in regions
+        surrounds the n-th generator in points and that the k-th
+        Voronoi vertex in vertices is the circumcenter of the k-th triangle
+        in self._simplices.  For each region n, we choose the first triangle
+        (=Voronoi vertex) in self._simplices and a vertex of that triangle
+        not equal to the center n. These determine a unique neighbor of that
+        triangle, which is then chosen as the second triangle. The second
+        triangle will have a unique vertex not equal to the current vertex or
+        the center. This determines a unique neighbor of the second triangle,
+        which is then chosen as the third triangle and so forth. We proceed
+        through all the triangles (=Voronoi vertices) belonging to the
+        generator in points and obtain a sorted version of the vertices
+        of its surrounding region.
+        """
+        if self._dim != 3:
+            raise TypeError("Only supported for three-dimensional point sets")
+        _voronoi.sort_vertices_of_regions(self._simplices, self.regions)
+
+    def _calculate_areas_3d(self):
+        self.sort_vertices_of_regions()
+        sizes = [len(region) for region in self.regions]
+        csizes = np.cumsum(sizes)
+        num_regions = csizes[-1]
+
+        # We create a set of triangles consisting of one point and two Voronoi
+        # vertices. The vertices of each triangle are adjacent in the sorted
+        # regions list.
+        point_indices = [i for i, size in enumerate(sizes)
+                         for j in range(size)]
+
+        nbrs1 = np.array([r for region in self.regions for r in region])
+
+        # The calculation of nbrs2 is a vectorized version of:
+        # np.array([r for region in self.regions for r in np.roll(region, 1)])
+        nbrs2 = np.roll(nbrs1, 1)
+        indices = np.roll(csizes, 1)
+        indices[0] = 0
+        nbrs2[indices] = nbrs1[csizes - 1]
+
+        # Normalize points and vertices.
+        pnormalized = (self.points - self.center) / self.radius
+        vnormalized = (self.vertices - self.center) / self.radius
+
+        # Create the complete set of triangles and calculate their solid angles
+        triangles = np.hstack([pnormalized[point_indices],
+                               vnormalized[nbrs1],
+                               vnormalized[nbrs2]
+                               ]).reshape((num_regions, 3, 3))
+        triangle_solid_angles = calculate_solid_angles(triangles)
+
+        # Sum the solid angles of the triangles in each region
+        solid_angles = np.cumsum(triangle_solid_angles)[csizes - 1]
+        solid_angles[1:] -= solid_angles[:-1]
+
+        # Get polygon areas using A = omega * r**2
+        return solid_angles * self.radius**2
+
+    def _calculate_areas_2d(self):
+        # Find start and end points of arcs
+        arcs = self.points[self._simplices] - self.center
+
+        # Calculate the angle subtended by arcs
+        cosine = np.einsum('ij,ij->i', arcs[:, 0], arcs[:, 1])
+        sine = np.abs(np.linalg.det(arcs))
+        theta = np.arctan2(sine, cosine)
+
+        # Get areas using A = r * theta
+        areas = self.radius * theta
+
+        # Correct arcs which go the wrong way (single-hemisphere inputs)
+        signs = np.sign(np.einsum('ij,ij->i', arcs[:, 0],
+                                              self.vertices - self.center))
+        indices = np.where(signs < 0)
+        areas[indices] = 2 * np.pi * self.radius - areas[indices]
+        return areas
+
+    def calculate_areas(self):
+        """Calculates the areas of the Voronoi regions.
+
+        For 2D point sets, the regions are circular arcs. The sum of the areas
+        is `2 * pi * radius`.
+
+        For 3D point sets, the regions are spherical polygons. The sum of the
+        areas is `4 * pi * radius**2`.
+
+        .. versionadded:: 1.5.0
+
+        Returns
+        -------
+        areas : double array of shape (npoints,)
+            The areas of the Voronoi regions.
+        """
+        if self._dim == 2:
+            return self._calculate_areas_2d()
+        elif self._dim == 3:
+            return self._calculate_areas_3d()
+        else:
+            raise TypeError("Only supported for 2D and 3D point sets")

venv/Lib/site-packages/scipy/spatial/_voronoi.pyi

@@ -0,0 +1,5 @@
+from typing import List
+
+import numpy as np
+
+def sort_vertices_of_regions(simplices: np.ndarray, regions: List[List[int]]) -> None: ...

venv/Lib/site-packages/scipy/spatial/ckdtree.pyi

@@ -0,0 +1 @@
+class cKDTree: ...

venv/Lib/site-packages/scipy/spatial/kdtree.py

@@ -0,0 +1,999 @@
+# Copyright Anne M. Archibald 2008
+# Released under the scipy license
+import numpy as np
+from heapq import heappush, heappop
+import scipy.sparse
+
+__all__ = ['minkowski_distance_p', 'minkowski_distance',
+           'distance_matrix',
+           'Rectangle', 'KDTree']
+
+
+def minkowski_distance_p(x, y, p=2):
+    """
+    Compute the pth power of the L**p distance between two arrays.
+
+    For efficiency, this function computes the L**p distance but does
+    not extract the pth root. If `p` is 1 or infinity, this is equal to
+    the actual L**p distance.
+
+    Parameters
+    ----------
+    x : (M, K) array_like
+        Input array.
+    y : (N, K) array_like
+        Input array.
+    p : float, 1 <= p <= infinity
+        Which Minkowski p-norm to use.
+
+    Examples
+    --------
+    >>> from scipy.spatial import minkowski_distance_p
+    >>> minkowski_distance_p([[0,0],[0,0]], [[1,1],[0,1]])
+    array([2, 1])
+
+    """
+    x = np.asarray(x)
+    y = np.asarray(y)
+
+    # Find smallest common datatype with float64 (return type of this function) - addresses #10262.
+    # Don't just cast to float64 for complex input case.
+    common_datatype = np.promote_types(np.promote_types(x.dtype, y.dtype), 'float64')
+
+    # Make sure x and y are NumPy arrays of correct datatype.
+    x = x.astype(common_datatype)
+    y = y.astype(common_datatype)
+
+    if p == np.inf:
+        return np.amax(np.abs(y-x), axis=-1)
+    elif p == 1:
+        return np.sum(np.abs(y-x), axis=-1)
+    else:
+        return np.sum(np.abs(y-x)**p, axis=-1)
+
+
+def minkowski_distance(x, y, p=2):
+    """
+    Compute the L**p distance between two arrays.
+
+    Parameters
+    ----------
+    x : (M, K) array_like
+        Input array.
+    y : (N, K) array_like
+        Input array.
+    p : float, 1 <= p <= infinity
+        Which Minkowski p-norm to use.
+
+    Examples
+    --------
+    >>> from scipy.spatial import minkowski_distance
+    >>> minkowski_distance([[0,0],[0,0]], [[1,1],[0,1]])
+    array([ 1.41421356,  1.        ])
+
+    """
+    x = np.asarray(x)
+    y = np.asarray(y)
+    if p == np.inf or p == 1:
+        return minkowski_distance_p(x, y, p)
+    else:
+        return minkowski_distance_p(x, y, p)**(1./p)
+
+
+class Rectangle(object):
+    """Hyperrectangle class.
+
+    Represents a Cartesian product of intervals.
+    """
+    def __init__(self, maxes, mins):
+        """Construct a hyperrectangle."""
+        self.maxes = np.maximum(maxes,mins).astype(float)
+        self.mins = np.minimum(maxes,mins).astype(float)
+        self.m, = self.maxes.shape
+
+    def __repr__(self):
+        return "<Rectangle %s>" % list(zip(self.mins, self.maxes))
+
+    def volume(self):
+        """Total volume."""
+        return np.prod(self.maxes-self.mins)
+
+    def split(self, d, split):
+        """
+        Produce two hyperrectangles by splitting.
+
+        In general, if you need to compute maximum and minimum
+        distances to the children, it can be done more efficiently
+        by updating the maximum and minimum distances to the parent.
+
+        Parameters
+        ----------
+        d : int
+            Axis to split hyperrectangle along.
+        split : float
+            Position along axis `d` to split at.
+
+        """
+        mid = np.copy(self.maxes)
+        mid[d] = split
+        less = Rectangle(self.mins, mid)
+        mid = np.copy(self.mins)
+        mid[d] = split
+        greater = Rectangle(mid, self.maxes)
+        return less, greater
+
+    def min_distance_point(self, x, p=2.):
+        """
+        Return the minimum distance between input and points in the hyperrectangle.
+
+        Parameters
+        ----------
+        x : array_like
+            Input.
+        p : float, optional
+            Input.
+
+        """
+        return minkowski_distance(0, np.maximum(0,np.maximum(self.mins-x,x-self.maxes)),p)
+
+    def max_distance_point(self, x, p=2.):
+        """
+        Return the maximum distance between input and points in the hyperrectangle.
+
+        Parameters
+        ----------
+        x : array_like
+            Input array.
+        p : float, optional
+            Input.
+
+        """
+        return minkowski_distance(0, np.maximum(self.maxes-x,x-self.mins),p)
+
+    def min_distance_rectangle(self, other, p=2.):
+        """
+        Compute the minimum distance between points in the two hyperrectangles.
+
+        Parameters
+        ----------
+        other : hyperrectangle
+            Input.
+        p : float
+            Input.
+
+        """
+        return minkowski_distance(0, np.maximum(0,np.maximum(self.mins-other.maxes,other.mins-self.maxes)),p)
+
+    def max_distance_rectangle(self, other, p=2.):
+        """
+        Compute the maximum distance between points in the two hyperrectangles.
+
+        Parameters
+        ----------
+        other : hyperrectangle
+            Input.
+        p : float, optional
+            Input.
+
+        """
+        return minkowski_distance(0, np.maximum(self.maxes-other.mins,other.maxes-self.mins),p)
+
+
+class KDTree(object):
+    """
+    kd-tree for quick nearest-neighbor lookup
+
+    This class provides an index into a set of k-D points which
+    can be used to rapidly look up the nearest neighbors of any point.
+
+    Parameters
+    ----------
+    data : (N,K) array_like
+        The data points to be indexed. This array is not copied, and
+        so modifying this data will result in bogus results.
+    leafsize : int, optional
+        The number of points at which the algorithm switches over to
+        brute-force.  Has to be positive.
+
+    Raises
+    ------
+    RuntimeError
+        The maximum recursion limit can be exceeded for large data
+        sets.  If this happens, either increase the value for the `leafsize`
+        parameter or increase the recursion limit by::
+
+            >>> import sys
+            >>> sys.setrecursionlimit(10000)
+
+    See Also
+    --------
+    cKDTree : Implementation of `KDTree` in Cython
+
+    Notes
+    -----
+    The algorithm used is described in Maneewongvatana and Mount 1999.
+    The general idea is that the kd-tree is a binary tree, each of whose
+    nodes represents an axis-aligned hyperrectangle. Each node specifies
+    an axis and splits the set of points based on whether their coordinate
+    along that axis is greater than or less than a particular value.
+
+    During construction, the axis and splitting point are chosen by the
+    "sliding midpoint" rule, which ensures that the cells do not all
+    become long and thin.
+
+    The tree can be queried for the r closest neighbors of any given point
+    (optionally returning only those within some maximum distance of the
+    point). It can also be queried, with a substantial gain in efficiency,
+    for the r approximate closest neighbors.
+
+    For large dimensions (20 is already large) do not expect this to run
+    significantly faster than brute force. High-dimensional nearest-neighbor
+    queries are a substantial open problem in computer science.
+
+    The tree also supports all-neighbors queries, both with arrays of points
+    and with other kd-trees. These do use a reasonably efficient algorithm,
+    but the kd-tree is not necessarily the best data structure for this
+    sort of calculation.
+
+    """
+    def __init__(self, data, leafsize=10):
+        self.data = np.asarray(data)
+        if self.data.dtype.kind == 'c':
+            raise TypeError("KDTree does not work with complex data")
+
+        self.n, self.m = np.shape(self.data)
+        self.leafsize = int(leafsize)
+        if self.leafsize < 1:
+            raise ValueError("leafsize must be at least 1")
+        self.maxes = np.amax(self.data,axis=0)
+        self.mins = np.amin(self.data,axis=0)
+
+        self.tree = self.__build(np.arange(self.n), self.maxes, self.mins)
+
+    class node(object):
+        def __lt__(self, other):
+            return id(self) < id(other)
+
+        def __gt__(self, other):
+            return id(self) > id(other)
+
+        def __le__(self, other):
+            return id(self) <= id(other)
+
+        def __ge__(self, other):
+            return id(self) >= id(other)
+
+        def __eq__(self, other):
+            return id(self) == id(other)
+
+    class leafnode(node):
+        def __init__(self, idx):
+            self.idx = idx
+            self.children = len(idx)
+
+    class innernode(node):
+        def __init__(self, split_dim, split, less, greater):
+            self.split_dim = split_dim
+            self.split = split
+            self.less = less
+            self.greater = greater
+            self.children = less.children+greater.children
+
+    def __build(self, idx, maxes, mins):
+        if len(idx) <= self.leafsize:
+            return KDTree.leafnode(idx)
+        else:
+            data = self.data[idx]
+            # maxes = np.amax(data,axis=0)
+            # mins = np.amin(data,axis=0)
+            d = np.argmax(maxes-mins)
+            maxval = maxes[d]
+            minval = mins[d]
+            if maxval == minval:
+                # all points are identical; warn user?
+                return KDTree.leafnode(idx)
+            data = data[:,d]
+
+            # sliding midpoint rule; see Maneewongvatana and Mount 1999
+            # for arguments that this is a good idea.
+            split = (maxval+minval)/2
+            less_idx = np.nonzero(data <= split)[0]
+            greater_idx = np.nonzero(data > split)[0]
+            if len(less_idx) == 0:
+                split = np.amin(data)
+                less_idx = np.nonzero(data <= split)[0]
+                greater_idx = np.nonzero(data > split)[0]
+            if len(greater_idx) == 0:
+                split = np.amax(data)
+                less_idx = np.nonzero(data < split)[0]
+                greater_idx = np.nonzero(data >= split)[0]
+            if len(less_idx) == 0:
+                # _still_ zero? all must have the same value
+                if not np.all(data == data[0]):
+                    raise ValueError("Troublesome data array: %s" % data)
+                split = data[0]
+                less_idx = np.arange(len(data)-1)
+                greater_idx = np.array([len(data)-1])
+
+            lessmaxes = np.copy(maxes)
+            lessmaxes[d] = split
+            greatermins = np.copy(mins)
+            greatermins[d] = split
+            return KDTree.innernode(d, split,
+                    self.__build(idx[less_idx],lessmaxes,mins),
+                    self.__build(idx[greater_idx],maxes,greatermins))
+
+    def __query(self, x, k=1, eps=0, p=2, distance_upper_bound=np.inf):
+
+        side_distances = np.maximum(0,np.maximum(x-self.maxes,self.mins-x))
+        if p != np.inf:
+            side_distances **= p
+            min_distance = np.sum(side_distances)
+        else:
+            min_distance = np.amax(side_distances)
+
+        # priority queue for chasing nodes
+        # entries are:
+        #  minimum distance between the cell and the target
+        #  distances between the nearest side of the cell and the target
+        #  the head node of the cell
+        q = [(min_distance,
+              tuple(side_distances),
+              self.tree)]
+        # priority queue for the nearest neighbors
+        # furthest known neighbor first
+        # entries are (-distance**p, i)
+        neighbors = []
+
+        if eps == 0:
+            epsfac = 1
+        elif p == np.inf:
+            epsfac = 1/(1+eps)
+        else:
+            epsfac = 1/(1+eps)**p
+
+        if p != np.inf and distance_upper_bound != np.inf:
+            distance_upper_bound = distance_upper_bound**p
+
+        while q:
+            min_distance, side_distances, node = heappop(q)
+            if isinstance(node, KDTree.leafnode):
+                # brute-force
+                data = self.data[node.idx]
+                ds = minkowski_distance_p(data,x[np.newaxis,:],p)
+                for i in range(len(ds)):
+                    if ds[i] < distance_upper_bound:
+                        if len(neighbors) == k:
+                            heappop(neighbors)
+                        heappush(neighbors, (-ds[i], node.idx[i]))
+                        if len(neighbors) == k:
+                            distance_upper_bound = -neighbors[0][0]
+            else:
+                # we don't push cells that are too far onto the queue at all,
+                # but since the distance_upper_bound decreases, we might get
+                # here even if the cell's too far
+                if min_distance > distance_upper_bound*epsfac:
+                    # since this is the nearest cell, we're done, bail out
+                    break
+                # compute minimum distances to the children and push them on
+                if x[node.split_dim] < node.split:
+                    near, far = node.less, node.greater
+                else:
+                    near, far = node.greater, node.less
+
+                # near child is at the same distance as the current node
+                heappush(q,(min_distance, side_distances, near))
+
+                # far child is further by an amount depending only
+                # on the split value
+                sd = list(side_distances)
+                if p == np.inf:
+                    min_distance = max(min_distance, abs(node.split-x[node.split_dim]))
+                elif p == 1:
+                    sd[node.split_dim] = np.abs(node.split-x[node.split_dim])
+                    min_distance = min_distance - side_distances[node.split_dim] + sd[node.split_dim]
+                else:
+                    sd[node.split_dim] = np.abs(node.split-x[node.split_dim])**p
+                    min_distance = min_distance - side_distances[node.split_dim] + sd[node.split_dim]
+
+                # far child might be too far, if so, don't bother pushing it
+                if min_distance <= distance_upper_bound*epsfac:
+                    heappush(q,(min_distance, tuple(sd), far))
+
+        if p == np.inf:
+            return sorted([(-d,i) for (d,i) in neighbors])
+        else:
+            return sorted([((-d)**(1./p),i) for (d,i) in neighbors])
+
+    def query(self, x, k=1, eps=0, p=2, distance_upper_bound=np.inf):
+        """
+        Query the kd-tree for nearest neighbors
+
+        Parameters
+        ----------
+        x : array_like, last dimension self.m
+            An array of points to query.
+        k : int, optional
+            The number of nearest neighbors to return.
+        eps : nonnegative float, optional
+            Return approximate nearest neighbors; the kth returned value
+            is guaranteed to be no further than (1+eps) times the
+            distance to the real kth nearest neighbor.
+        p : float, 1<=p<=infinity, optional
+            Which Minkowski p-norm to use.
+            1 is the sum-of-absolute-values "Manhattan" distance
+            2 is the usual Euclidean distance
+            infinity is the maximum-coordinate-difference distance
+        distance_upper_bound : nonnegative float, optional
+            Return only neighbors within this distance. This is used to prune
+            tree searches, so if you are doing a series of nearest-neighbor
+            queries, it may help to supply the distance to the nearest neighbor
+            of the most recent point.
+
+        Returns
+        -------
+        d : float or array of floats
+            The distances to the nearest neighbors.
+            If x has shape tuple+(self.m,), then d has shape tuple if
+            k is one, or tuple+(k,) if k is larger than one. Missing
+            neighbors (e.g. when k > n or distance_upper_bound is
+            given) are indicated with infinite distances.  If k is None,
+            then d is an object array of shape tuple, containing lists
+            of distances. In either case the hits are sorted by distance
+            (nearest first).
+        i : integer or array of integers
+            The locations of the neighbors in self.data. i is the same
+            shape as d.
+
+        Examples
+        --------
+        >>> from scipy import spatial
+        >>> x, y = np.mgrid[0:5, 2:8]
+        >>> tree = spatial.KDTree(list(zip(x.ravel(), y.ravel())))
+        >>> tree.data
+        array([[0, 2],
+               [0, 3],
+               [0, 4],
+               [0, 5],
+               [0, 6],
+               [0, 7],
+               [1, 2],
+               [1, 3],
+               [1, 4],
+               [1, 5],
+               [1, 6],
+               [1, 7],
+               [2, 2],
+               [2, 3],
+               [2, 4],
+               [2, 5],
+               [2, 6],
+               [2, 7],
+               [3, 2],
+               [3, 3],
+               [3, 4],
+               [3, 5],
+               [3, 6],
+               [3, 7],
+               [4, 2],
+               [4, 3],
+               [4, 4],
+               [4, 5],
+               [4, 6],
+               [4, 7]])
+        >>> pts = np.array([[0, 0], [2.1, 2.9]])
+        >>> tree.query(pts)
+        (array([ 2.        ,  0.14142136]), array([ 0, 13]))
+        >>> tree.query(pts[0])
+        (2.0, 0)
+
+        """
+        x = np.asarray(x)
+        if x.dtype.kind == 'c':
+            raise TypeError("KDTree does not work with complex data")
+        if np.shape(x)[-1] != self.m:
+            raise ValueError("x must consist of vectors of length %d but has shape %s" % (self.m, np.shape(x)))
+        if p < 1:
+            raise ValueError("Only p-norms with 1<=p<=infinity permitted")
+        retshape = np.shape(x)[:-1]
+        if retshape != ():
+            if k is None:
+                dd = np.empty(retshape,dtype=object)
+                ii = np.empty(retshape,dtype=object)
+            elif k > 1:
+                dd = np.empty(retshape+(k,),dtype=float)
+                dd.fill(np.inf)
+                ii = np.empty(retshape+(k,),dtype=int)
+                ii.fill(self.n)
+            elif k == 1:
+                dd = np.empty(retshape,dtype=float)
+                dd.fill(np.inf)
+                ii = np.empty(retshape,dtype=int)
+                ii.fill(self.n)
+            else:
+                raise ValueError("Requested %s nearest neighbors; acceptable numbers are integers greater than or equal to one, or None")
+            for c in np.ndindex(retshape):
+                hits = self.__query(x[c], k=k, eps=eps, p=p, distance_upper_bound=distance_upper_bound)
+                if k is None:
+                    dd[c] = [d for (d,i) in hits]
+                    ii[c] = [i for (d,i) in hits]
+                elif k > 1:
+                    for j in range(len(hits)):
+                        dd[c+(j,)], ii[c+(j,)] = hits[j]
+                elif k == 1:
+                    if len(hits) > 0:
+                        dd[c], ii[c] = hits[0]
+                    else:
+                        dd[c] = np.inf
+                        ii[c] = self.n
+            return dd, ii
+        else:
+            hits = self.__query(x, k=k, eps=eps, p=p, distance_upper_bound=distance_upper_bound)
+            if k is None:
+                return [d for (d,i) in hits], [i for (d,i) in hits]
+            elif k == 1:
+                if len(hits) > 0:
+                    return hits[0]
+                else:
+                    return np.inf, self.n
+            elif k > 1:
+                dd = np.empty(k,dtype=float)
+                dd.fill(np.inf)
+                ii = np.empty(k,dtype=int)
+                ii.fill(self.n)
+                for j in range(len(hits)):
+                    dd[j], ii[j] = hits[j]
+                return dd, ii
+            else:
+                raise ValueError("Requested %s nearest neighbors; acceptable numbers are integers greater than or equal to one, or None")
+
+    def __query_ball_point(self, x, r, p=2., eps=0):
+        R = Rectangle(self.maxes, self.mins)
+
+        def traverse_checking(node, rect):
+            if rect.min_distance_point(x, p) > r / (1. + eps):
+                return []
+            elif rect.max_distance_point(x, p) < r * (1. + eps):
+                return traverse_no_checking(node)
+            elif isinstance(node, KDTree.leafnode):
+                d = self.data[node.idx]
+                return node.idx[minkowski_distance(d, x, p) <= r].tolist()
+            else:
+                less, greater = rect.split(node.split_dim, node.split)
+                return traverse_checking(node.less, less) + \
+                       traverse_checking(node.greater, greater)
+
+        def traverse_no_checking(node):
+            if isinstance(node, KDTree.leafnode):
+                return node.idx.tolist()
+            else:
+                return traverse_no_checking(node.less) + \
+                       traverse_no_checking(node.greater)
+
+        return traverse_checking(self.tree, R)
+
+    def query_ball_point(self, x, r, p=2., eps=0):
+        """Find all points within distance r of point(s) x.
+
+        Parameters
+        ----------
+        x : array_like, shape tuple + (self.m,)
+            The point or points to search for neighbors of.
+        r : positive float
+            The radius of points to return.
+        p : float, optional
+            Which Minkowski p-norm to use.  Should be in the range [1, inf].
+        eps : nonnegative float, optional
+            Approximate search. Branches of the tree are not explored if their
+            nearest points are further than ``r / (1 + eps)``, and branches are
+            added in bulk if their furthest points are nearer than
+            ``r * (1 + eps)``.
+
+        Returns
+        -------
+        results : list or array of lists
+            If `x` is a single point, returns a list of the indices of the
+            neighbors of `x`. If `x` is an array of points, returns an object
+            array of shape tuple containing lists of neighbors.
+
+        Notes
+        -----
+        If you have many points whose neighbors you want to find, you may save
+        substantial amounts of time by putting them in a KDTree and using
+        query_ball_tree.
+
+        Examples
+        --------
+        >>> from scipy import spatial
+        >>> x, y = np.mgrid[0:5, 0:5]
+        >>> points = np.c_[x.ravel(), y.ravel()]
+        >>> tree = spatial.KDTree(points)
+        >>> tree.query_ball_point([2, 0], 1)
+        [5, 10, 11, 15]
+
+        Query multiple points and plot the results:
+
+        >>> import matplotlib.pyplot as plt
+        >>> points = np.asarray(points)
+        >>> plt.plot(points[:,0], points[:,1], '.')
+        >>> for results in tree.query_ball_point(([2, 0], [3, 3]), 1):
+        ...     nearby_points = points[results]
+        ...     plt.plot(nearby_points[:,0], nearby_points[:,1], 'o')
+        >>> plt.margins(0.1, 0.1)
+        >>> plt.show()
+
+        """
+        x = np.asarray(x)
+        if x.dtype.kind == 'c':
+            raise TypeError("KDTree does not work with complex data")
+        if x.shape[-1] != self.m:
+            raise ValueError("Searching for a %d-dimensional point in a "
+                             "%d-dimensional KDTree" % (x.shape[-1], self.m))
+        if len(x.shape) == 1:
+            return self.__query_ball_point(x, r, p, eps)
+        else:
+            retshape = x.shape[:-1]
+            result = np.empty(retshape, dtype=object)
+            for c in np.ndindex(retshape):
+                result[c] = self.__query_ball_point(x[c], r, p=p, eps=eps)
+            return result
+
+    def query_ball_tree(self, other, r, p=2., eps=0):
+        """Find all pairs of points whose distance is at most r
+
+        Parameters
+        ----------
+        other : KDTree instance
+            The tree containing points to search against.
+        r : float
+            The maximum distance, has to be positive.
+        p : float, optional
+            Which Minkowski norm to use.  `p` has to meet the condition
+            ``1 <= p <= infinity``.
+        eps : float, optional
+            Approximate search.  Branches of the tree are not explored
+            if their nearest points are further than ``r/(1+eps)``, and
+            branches are added in bulk if their furthest points are nearer
+            than ``r * (1+eps)``.  `eps` has to be non-negative.
+
+        Returns
+        -------
+        results : list of lists
+            For each element ``self.data[i]`` of this tree, ``results[i]`` is a
+            list of the indices of its neighbors in ``other.data``.
+
+        """
+        results = [[] for i in range(self.n)]
+
+        def traverse_checking(node1, rect1, node2, rect2):
+            if rect1.min_distance_rectangle(rect2, p) > r/(1.+eps):
+                return
+            elif rect1.max_distance_rectangle(rect2, p) < r*(1.+eps):
+                traverse_no_checking(node1, node2)
+            elif isinstance(node1, KDTree.leafnode):
+                if isinstance(node2, KDTree.leafnode):
+                    d = other.data[node2.idx]
+                    for i in node1.idx:
+                        results[i] += node2.idx[minkowski_distance(d,self.data[i],p) <= r].tolist()
+                else:
+                    less, greater = rect2.split(node2.split_dim, node2.split)
+                    traverse_checking(node1,rect1,node2.less,less)
+                    traverse_checking(node1,rect1,node2.greater,greater)
+            elif isinstance(node2, KDTree.leafnode):
+                less, greater = rect1.split(node1.split_dim, node1.split)
+                traverse_checking(node1.less,less,node2,rect2)
+                traverse_checking(node1.greater,greater,node2,rect2)
+            else:
+                less1, greater1 = rect1.split(node1.split_dim, node1.split)
+                less2, greater2 = rect2.split(node2.split_dim, node2.split)
+                traverse_checking(node1.less,less1,node2.less,less2)
+                traverse_checking(node1.less,less1,node2.greater,greater2)
+                traverse_checking(node1.greater,greater1,node2.less,less2)
+                traverse_checking(node1.greater,greater1,node2.greater,greater2)
+
+        def traverse_no_checking(node1, node2):
+            if isinstance(node1, KDTree.leafnode):
+                if isinstance(node2, KDTree.leafnode):
+                    for i in node1.idx:
+                        results[i] += node2.idx.tolist()
+                else:
+                    traverse_no_checking(node1, node2.less)
+                    traverse_no_checking(node1, node2.greater)
+            else:
+                traverse_no_checking(node1.less, node2)
+                traverse_no_checking(node1.greater, node2)
+
+        traverse_checking(self.tree, Rectangle(self.maxes, self.mins),
+                          other.tree, Rectangle(other.maxes, other.mins))
+        return results
+
+    def query_pairs(self, r, p=2., eps=0):
+        """
+        Find all pairs of points within a distance.
+
+        Parameters
+        ----------
+        r : positive float
+            The maximum distance.
+        p : float, optional
+            Which Minkowski norm to use.  `p` has to meet the condition
+            ``1 <= p <= infinity``.
+        eps : float, optional
+            Approximate search.  Branches of the tree are not explored
+            if their nearest points are further than ``r/(1+eps)``, and
+            branches are added in bulk if their furthest points are nearer
+            than ``r * (1+eps)``.  `eps` has to be non-negative.
+
+        Returns
+        -------
+        results : set
+            Set of pairs ``(i,j)``, with ``i < j``, for which the corresponding
+            positions are close.
+
+        """
+        results = set()
+
+        def traverse_checking(node1, rect1, node2, rect2):
+            if rect1.min_distance_rectangle(rect2, p) > r/(1.+eps):
+                return
+            elif rect1.max_distance_rectangle(rect2, p) < r*(1.+eps):
+                traverse_no_checking(node1, node2)
+            elif isinstance(node1, KDTree.leafnode):
+                if isinstance(node2, KDTree.leafnode):
+                    # Special care to avoid duplicate pairs
+                    if id(node1) == id(node2):
+                        d = self.data[node2.idx]
+                        for i in node1.idx:
+                            for j in node2.idx[minkowski_distance(d,self.data[i],p) <= r]:
+                                if i < j:
+                                    results.add((i,j))
+                    else:
+                        d = self.data[node2.idx]
+                        for i in node1.idx:
+                            for j in node2.idx[minkowski_distance(d,self.data[i],p) <= r]:
+                                if i < j:
+                                    results.add((i,j))
+                                elif j < i:
+                                    results.add((j,i))
+                else:
+                    less, greater = rect2.split(node2.split_dim, node2.split)
+                    traverse_checking(node1,rect1,node2.less,less)
+                    traverse_checking(node1,rect1,node2.greater,greater)
+            elif isinstance(node2, KDTree.leafnode):
+                less, greater = rect1.split(node1.split_dim, node1.split)
+                traverse_checking(node1.less,less,node2,rect2)
+                traverse_checking(node1.greater,greater,node2,rect2)
+            else:
+                less1, greater1 = rect1.split(node1.split_dim, node1.split)
+                less2, greater2 = rect2.split(node2.split_dim, node2.split)
+                traverse_checking(node1.less,less1,node2.less,less2)
+                traverse_checking(node1.less,less1,node2.greater,greater2)
+
+                # Avoid traversing (node1.less, node2.greater) and
+                # (node1.greater, node2.less) (it's the same node pair twice
+                # over, which is the source of the complication in the
+                # original KDTree.query_pairs)
+                if id(node1) != id(node2):
+                    traverse_checking(node1.greater,greater1,node2.less,less2)
+
+                traverse_checking(node1.greater,greater1,node2.greater,greater2)
+
+        def traverse_no_checking(node1, node2):
+            if isinstance(node1, KDTree.leafnode):
+                if isinstance(node2, KDTree.leafnode):
+                    # Special care to avoid duplicate pairs
+                    if id(node1) == id(node2):
+                        for i in node1.idx:
+                            for j in node2.idx:
+                                if i < j:
+                                    results.add((i,j))
+                    else:
+                        for i in node1.idx:
+                            for j in node2.idx:
+                                if i < j:
+                                    results.add((i,j))
+                                elif j < i:
+                                    results.add((j,i))
+                else:
+                    traverse_no_checking(node1, node2.less)
+                    traverse_no_checking(node1, node2.greater)
+            else:
+                # Avoid traversing (node1.less, node2.greater) and
+                # (node1.greater, node2.less) (it's the same node pair twice
+                # over, which is the source of the complication in the
+                # original KDTree.query_pairs)
+                if id(node1) == id(node2):
+                    traverse_no_checking(node1.less, node2.less)
+                    traverse_no_checking(node1.less, node2.greater)
+                    traverse_no_checking(node1.greater, node2.greater)
+                else:
+                    traverse_no_checking(node1.less, node2)
+                    traverse_no_checking(node1.greater, node2)
+
+        traverse_checking(self.tree, Rectangle(self.maxes, self.mins),
+                          self.tree, Rectangle(self.maxes, self.mins))
+        return results
+
+    def count_neighbors(self, other, r, p=2.):
+        """
+        Count how many nearby pairs can be formed.
+
+        Count the number of pairs (x1,x2) can be formed, with x1 drawn
+        from self and x2 drawn from ``other``, and where
+        ``distance(x1, x2, p) <= r``.
+        This is the "two-point correlation" described in Gray and Moore 2000,
+        "N-body problems in statistical learning", and the code here is based
+        on their algorithm.
+
+        Parameters
+        ----------
+        other : KDTree instance
+            The other tree to draw points from.
+        r : float or one-dimensional array of floats
+            The radius to produce a count for. Multiple radii are searched with
+            a single tree traversal.
+        p : float, 1<=p<=infinity, optional
+            Which Minkowski p-norm to use
+
+        Returns
+        -------
+        result : int or 1-D array of ints
+            The number of pairs. Note that this is internally stored in a numpy
+            int, and so may overflow if very large (2e9).
+
+        """
+        def traverse(node1, rect1, node2, rect2, idx):
+            min_r = rect1.min_distance_rectangle(rect2,p)
+            max_r = rect1.max_distance_rectangle(rect2,p)
+            c_greater = r[idx] > max_r
+            result[idx[c_greater]] += node1.children*node2.children
+            idx = idx[(min_r <= r[idx]) & (r[idx] <= max_r)]
+            if len(idx) == 0:
+                return
+
+            if isinstance(node1,KDTree.leafnode):
+                if isinstance(node2,KDTree.leafnode):
+                    ds = minkowski_distance(self.data[node1.idx][:,np.newaxis,:],
+                                  other.data[node2.idx][np.newaxis,:,:],
+                                  p).ravel()
+                    ds.sort()
+                    result[idx] += np.searchsorted(ds,r[idx],side='right')
+                else:
+                    less, greater = rect2.split(node2.split_dim, node2.split)
+                    traverse(node1, rect1, node2.less, less, idx)
+                    traverse(node1, rect1, node2.greater, greater, idx)
+            else:
+                if isinstance(node2,KDTree.leafnode):
+                    less, greater = rect1.split(node1.split_dim, node1.split)
+                    traverse(node1.less, less, node2, rect2, idx)
+                    traverse(node1.greater, greater, node2, rect2, idx)
+                else:
+                    less1, greater1 = rect1.split(node1.split_dim, node1.split)
+                    less2, greater2 = rect2.split(node2.split_dim, node2.split)
+                    traverse(node1.less,less1,node2.less,less2,idx)
+                    traverse(node1.less,less1,node2.greater,greater2,idx)
+                    traverse(node1.greater,greater1,node2.less,less2,idx)
+                    traverse(node1.greater,greater1,node2.greater,greater2,idx)
+
+        R1 = Rectangle(self.maxes, self.mins)
+        R2 = Rectangle(other.maxes, other.mins)
+        if np.shape(r) == ():
+            r = np.array([r])
+            result = np.zeros(1,dtype=int)
+            traverse(self.tree, R1, other.tree, R2, np.arange(1))
+            return result[0]
+        elif len(np.shape(r)) == 1:
+            r = np.asarray(r)
+            n, = r.shape
+            result = np.zeros(n,dtype=int)
+            traverse(self.tree, R1, other.tree, R2, np.arange(n))
+            return result
+        else:
+            raise ValueError("r must be either a single value or a one-dimensional array of values")
+
+    def sparse_distance_matrix(self, other, max_distance, p=2.):
+        """
+        Compute a sparse distance matrix
+
+        Computes a distance matrix between two KDTrees, leaving as zero
+        any distance greater than max_distance.
+
+        Parameters
+        ----------
+        other : KDTree
+
+        max_distance : positive float
+
+        p : float, optional
+
+        Returns
+        -------
+        result : dok_matrix
+            Sparse matrix representing the results in "dictionary of keys" format.
+
+        """
+        result = scipy.sparse.dok_matrix((self.n,other.n))
+
+        def traverse(node1, rect1, node2, rect2):
+            if rect1.min_distance_rectangle(rect2, p) > max_distance:
+                return
+            elif isinstance(node1, KDTree.leafnode):
+                if isinstance(node2, KDTree.leafnode):
+                    for i in node1.idx:
+                        for j in node2.idx:
+                            d = minkowski_distance(self.data[i],other.data[j],p)
+                            if d <= max_distance:
+                                result[i,j] = d
+                else:
+                    less, greater = rect2.split(node2.split_dim, node2.split)
+                    traverse(node1,rect1,node2.less,less)
+                    traverse(node1,rect1,node2.greater,greater)
+            elif isinstance(node2, KDTree.leafnode):
+                less, greater = rect1.split(node1.split_dim, node1.split)
+                traverse(node1.less,less,node2,rect2)
+                traverse(node1.greater,greater,node2,rect2)
+            else:
+                less1, greater1 = rect1.split(node1.split_dim, node1.split)
+                less2, greater2 = rect2.split(node2.split_dim, node2.split)
+                traverse(node1.less,less1,node2.less,less2)
+                traverse(node1.less,less1,node2.greater,greater2)
+                traverse(node1.greater,greater1,node2.less,less2)
+                traverse(node1.greater,greater1,node2.greater,greater2)
+        traverse(self.tree, Rectangle(self.maxes, self.mins),
+                 other.tree, Rectangle(other.maxes, other.mins))
+
+        return result
+
+
+def distance_matrix(x, y, p=2, threshold=1000000):
+    """
+    Compute the distance matrix.
+
+    Returns the matrix of all pair-wise distances.
+
+    Parameters
+    ----------
+    x : (M, K) array_like
+        Matrix of M vectors in K dimensions.
+    y : (N, K) array_like
+        Matrix of N vectors in K dimensions.
+    p : float, 1 <= p <= infinity
+        Which Minkowski p-norm to use.
+    threshold : positive int
+        If ``M * N * K`` > `threshold`, algorithm uses a Python loop instead
+        of large temporary arrays.
+
+    Returns
+    -------
+    result : (M, N) ndarray
+        Matrix containing the distance from every vector in `x` to every vector
+        in `y`.
+
+    Examples
+    --------
+    >>> from scipy.spatial import distance_matrix
+    >>> distance_matrix([[0,0],[0,1]], [[1,0],[1,1]])
+    array([[ 1.        ,  1.41421356],
+           [ 1.41421356,  1.        ]])
+
+    """
+
+    x = np.asarray(x)
+    m, k = x.shape
+    y = np.asarray(y)
+    n, kk = y.shape
+
+    if k != kk:
+        raise ValueError("x contains %d-dimensional vectors but y contains %d-dimensional vectors" % (k, kk))
+
+    if m*n*k <= threshold:
+        return minkowski_distance(x[:,np.newaxis,:],y[np.newaxis,:,:],p)
+    else:
+        result = np.empty((m,n),dtype=float)  # FIXME: figure out the best dtype
+        if m < n:
+            for i in range(m):
+                result[i,:] = minkowski_distance(x[i],y,p)
+        else:
+            for j in range(n):
+                result[:,j] = minkowski_distance(x,y[j],p)
+        return result

venv/Lib/site-packages/scipy/spatial/qhull_src/COPYING.txt

@@ -0,0 +1,38 @@
+                    Qhull, Copyright (c) 1993-2019
+                    
+                            C.B. Barber
+                           Arlington, MA 
+                          
+                               and
+
+       The National Science and Technology Research Center for
+        Computation and Visualization of Geometric Structures
+                        (The Geometry Center)
+                       University of Minnesota
+
+                       email: qhull@qhull.org
+
+This software includes Qhull from C.B. Barber and The Geometry Center.  
+Qhull is copyrighted as noted above.  Qhull is free software and may 
+be obtained via http from www.qhull.org.  It may be freely copied, modified, 
+and redistributed under the following conditions:
+
+1. All copyright notices must remain intact in all files.
+
+2. A copy of this text file must be distributed along with any copies 
+   of Qhull that you redistribute; this includes copies that you have 
+   modified, or copies of programs or other software products that 
+   include Qhull.
+
+3. If you modify Qhull, you must include a notice giving the
+   name of the person performing the modification, the date of
+   modification, and the reason for such modification.
+
+4. When distributing modified versions of Qhull, or other software 
+   products that include Qhull, you must provide notice that the original 
+   source code may be obtained as noted above.
+
+5. There is no warranty or other guarantee of fitness for Qhull, it is 
+   provided solely "as is".  Bug reports or fixes may be sent to 
+   qhull_bug@qhull.org; the authors may or may not act on them as 
+   they desire.

venv/Lib/site-packages/scipy/spatial/setup.py

@@ -0,0 +1,93 @@
+from os.path import join, dirname
+import glob
+
+
+def configuration(parent_package='', top_path=None):
+    from numpy.distutils.misc_util import Configuration, get_numpy_include_dirs
+    from numpy.distutils.misc_util import get_info as get_misc_info
+    from scipy._build_utils.system_info import get_info
+    from scipy._build_utils import combine_dict, uses_blas64
+    from scipy._build_utils.compiler_helper import set_cxx_flags_hook
+    from distutils.sysconfig import get_python_inc
+
+    config = Configuration('spatial', parent_package, top_path)
+
+    config.add_data_dir('tests')
+
+    # spatial.transform
+    config.add_subpackage('transform')
+
+    # qhull
+    qhull_src = sorted(glob.glob(join(dirname(__file__), 'qhull_src',
+                                    'src', '*.c')))
+
+    inc_dirs = [get_python_inc()]
+    if inc_dirs[0] != get_python_inc(plat_specific=1):
+        inc_dirs.append(get_python_inc(plat_specific=1))
+    inc_dirs.append(get_numpy_include_dirs())
+    inc_dirs.append(join(dirname(dirname(__file__)), '_lib'))
+    inc_dirs.append(join(dirname(dirname(__file__)), '_build_utils', 'src'))
+
+    if uses_blas64():
+        lapack_opt = get_info('lapack_ilp64_opt')
+    else:
+        lapack_opt = get_info('lapack_opt')
+
+    cfg = combine_dict(lapack_opt, include_dirs=inc_dirs)
+    config.add_extension('qhull',
+                         sources=['qhull.c', 'qhull_misc.c'] + qhull_src,
+                         **cfg)
+
+    # cKDTree
+    ckdtree_src = ['query.cxx',
+                   'build.cxx',
+                   'query_pairs.cxx',
+                   'count_neighbors.cxx',
+                   'query_ball_point.cxx',
+                   'query_ball_tree.cxx',
+                   'sparse_distances.cxx']
+
+    ckdtree_src = [join('ckdtree', 'src', x) for x in ckdtree_src]
+
+    ckdtree_headers = ['ckdtree_decl.h',
+                       'coo_entries.h',
+                       'distance_base.h',
+                       'distance.h',
+                       'ordered_pair.h',
+                       'partial_sort.h',
+                       'rectangle.h']
+
+    ckdtree_headers = [join('ckdtree', 'src', x) for x in ckdtree_headers]
+
+    ckdtree_dep = ['ckdtree.cxx'] + ckdtree_headers + ckdtree_src
+    ext = config.add_extension('ckdtree',
+                         sources=['ckdtree.cxx'] + ckdtree_src,
+                         depends=ckdtree_dep,
+                         include_dirs=inc_dirs + [join('ckdtree', 'src')])
+    ext._pre_build_hook = set_cxx_flags_hook
+
+    # _distance_wrap
+    config.add_extension('_distance_wrap',
+                         sources=[join('src', 'distance_wrap.c')],
+                         depends=[join('src', 'distance_impl.h')],
+                         include_dirs=[get_numpy_include_dirs()],
+                         extra_info=get_misc_info("npymath"))
+
+    config.add_extension('_voronoi',
+                         sources=['_voronoi.c'])
+
+    config.add_extension('_hausdorff',
+                         sources=['_hausdorff.c'])
+
+    # Add license files
+    config.add_data_files('qhull_src/COPYING.txt')
+
+    # Type stubs
+    config.add_data_files('*.pyi')
+
+    return config
+
+
+if __name__ == '__main__':
+    from numpy.distutils.core import setup
+    setup(**configuration(top_path='').todict())

venv/Lib/site-packages/scipy/spatial/tests/data/cdist-X1.txt

@@ -0,0 +1,10 @@
+1.147593763490969421e-01 8.926156143344999849e-01 1.437758624645746330e-02 1.803435962879929022e-02 5.533046214065578949e-01 5.554315640747428118e-01 4.497546637814608950e-02 4.438089247948049376e-01 7.984582810220538507e-01 2.752880789161644692e-01 1.344667112315823809e-01 9.230479561452992199e-01 6.040471462941819913e-01 3.797251652770228247e-01 4.316042735592399149e-01 5.312356915348823705e-01 4.348143005129563310e-01 3.111531488508799681e-01 9.531194313908697424e-04 8.212995023500069269e-02 6.689953269869852726e-01 9.914864535288493430e-01 8.037556036341153565e-01
+9.608925123801395074e-01 2.974451233678974127e-01 9.001110330654185088e-01 5.824163330415995654e-01 7.308574928293812834e-01 2.276154562412870952e-01 7.306791076039623745e-01 8.677244866905511333e-01 9.160806456176984192e-01 6.157216959991280714e-01 5.149053524695440531e-01 3.056427344890983999e-01 9.790557366933895223e-01 4.484995861076724877e-01 4.776550391081165747e-01 7.210436977670631187e-01 9.136399501661039979e-01 4.260275733550000776e-02 5.943900041968954717e-01 3.864571606342745991e-01 9.442027665110838131e-01 4.779949058608601309e-02 6.107551944250865228e-01
+3.297286578103622023e-01 5.980207401936733502e-01 3.673301293561567205e-01 2.585830520887681949e-01 4.660558746104259686e-01 6.083795956610364986e-01 4.535206368070313632e-01 6.873989778785424276e-01 5.130152688495458468e-01 7.665877846542720198e-01 3.444402973525138023e-01 3.583658123644906102e-02 7.924818220986856732e-01 8.746685720522412444e-01 3.010105569182431884e-01 6.012239357385538163e-01 6.233737362204671006e-01 4.830438698668915176e-01 2.317286885842551047e-02 7.585989958123050547e-01 7.108257632278830451e-01 1.551024884178199281e-01 2.665485998155288083e-01
+2.456278068903017253e-02 4.148739837711815648e-01 1.986372227934196655e-01 6.920408530298168825e-01 1.003067576685774398e-01 7.421560456480125190e-01 1.808453980608998313e-01 4.251297882537475870e-01 6.773002683522370004e-01 4.084108792570182445e-01 7.462888013191590897e-01 8.069930220529277776e-01 9.211110587681808903e-01 4.141491046181076108e-01 7.486318689260342829e-01 9.515405507589296263e-01 4.634288892577109742e-03 8.027593488166355762e-01 3.010346805217798405e-01 8.663248877242523127e-01 2.479968181181605447e-01 5.619851096054278017e-01 3.903886764590250857e-01
+7.122019976035700584e-01 6.188878051047785878e-01 7.290897087051201320e-01 6.334802157757637442e-01 5.523084734954342156e-01 5.614937129563645213e-01 2.496741051791574462e-01 5.972227939599233926e-01 1.786590597761109622e-01 2.609525984850900038e-01 7.210438943286010538e-01 2.211429064605652250e-01 9.140497572472672250e-02 1.430242193668443962e-01 7.856446942916397447e-01 4.635256358156553125e-01 5.278744289813760426e-01 3.702808015407184072e-01 5.527073830480792038e-01 6.370732917599846168e-01 9.953487928925482953e-01 3.021789770611936765e-01 3.354901923998221402e-02
+6.509638560895427695e-01 8.387598220902757751e-01 7.761375971745763103e-01 1.481627639227802717e-01 3.529474982902305324e-01 4.883093646287851586e-01 9.652923033658690199e-01 9.500680513565308294e-01 3.061885005078281985e-01 7.271902818906019750e-01 2.358962978196710303e-03 7.359889703223099211e-01 8.988893768074724955e-01 4.135279653937307121e-02 8.516441856688283796e-01 4.889597623270667270e-01 5.575909822114655245e-01 9.010853652261575641e-01 2.912844516556202246e-01 9.088759383368658629e-01 8.104351227460024898e-01 8.080695436776826890e-01 1.430530913253185155e-01
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+6.527328264244687261e-01 8.289483383553154505e-01 9.179741348282299818e-01 1.065639864466713105e-01 6.253616929058514184e-01 5.927750325266062381e-01 3.039157425463192563e-01 2.452766763359194302e-01 6.514027700704632107e-01 5.529218485487964463e-01 4.941158239308394151e-01 6.605306467722642516e-01 2.273688037050677346e-01 4.282616592244774534e-01 2.956128257930247250e-01 1.154803628237965896e-01 9.228220410235263849e-01 6.663525307676617659e-01 1.908852615936970087e-01 9.921383408926374159e-01 4.988716450388516188e-01 1.014900352736023414e-01 3.363930180244284474e-01
+2.914369076275757919e-01 5.196673601143533272e-01 7.420144907858341465e-01 1.768984185504740569e-01 5.296766993228564369e-01 5.922023566159900776e-01 5.965161262020234334e-01 3.810272333046110793e-01 8.368797246118340194e-01 7.896422363801189892e-01 9.655797561098209414e-01 4.430034032346981121e-01 2.780869795706976122e-01 3.047310845416009162e-01 8.051138863500326703e-01 6.731468634690835895e-01 4.743383036815584930e-01 9.530709614322225853e-01 7.753587619850917934e-01 2.801137109357491051e-01 6.182543660889736614e-01 5.005218857766725593e-01 9.071447804755052857e-01
+2.075071644012620453e-01 4.834950086973934802e-01 3.037011473860764532e-01 6.476084284887700937e-01 8.107195771564194020e-01 7.869075869075803364e-01 6.851234019375299633e-01 3.544187468104398331e-02 4.847673235908021017e-01 5.690262846164507726e-01 1.663354142616256803e-01 9.692796809752548537e-01 4.133441725866372485e-01 6.729167604487583665e-01 3.998813427407297283e-01 8.272617414104491695e-01 2.129248316324727774e-01 6.517004761357130249e-01 7.363013506605019520e-01 4.072375306356985636e-01 4.463336683526665238e-01 5.485059309728204102e-01 1.981745754527846071e-01

venv/Lib/site-packages/scipy/spatial/tests/data/cdist-X2.txt

@@ -0,0 +1,20 @@
+7.680465556300619667e-02 4.675022344069014180e-01 8.955498989131543963e-01 3.816236071436276411e-01 1.109030077070989329e-01 2.318928815459808668e-02 7.477394240984251983e-01 1.202289789304434864e-01 8.007290497575981769e-01 6.795195698871731027e-01 6.568225762396605605e-01 2.231475263228478445e-01 7.064624077661341151e-02 1.081656666815267176e-02 1.592069359090128033e-01 1.363392203645097389e-01 9.277020735447568667e-01 8.103136564528209407e-01 5.229467676276455812e-02 7.708020259874025504e-01 6.527954747473352359e-02 5.516397414886525796e-01 3.653371861367954443e-01
+8.144399106025798085e-01 7.731852525462976633e-01 6.909477620673205589e-01 9.696063817000286633e-01 4.297887511677249694e-01 6.989600553425188156e-01 7.310201335033380543e-01 3.135256147868910048e-01 5.715578037275241829e-01 3.935000744675094531e-01 2.057715781268398825e-01 5.892508589665171881e-01 8.512951599236765476e-01 9.569808799061578775e-01 6.164885878024699561e-01 4.714185430004367294e-01 6.128831737628155363e-01 6.641799309623502845e-01 6.001985185338730711e-01 4.231922889723856995e-01 7.605249308075449077e-01 1.064530958018087281e-01 6.306470691957204444e-01
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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-chebyshev-ml.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-cityblock-ml.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-correlation-ml.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-cosine-ml.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-double-inp.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-euclidean-ml.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-hamming-ml.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-jaccard-ml.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-jensenshannon-ml.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-minkowski-3.2-ml.txt

@@ -0,0 +1 @@
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venv/Lib/site-packages/scipy/spatial/tests/data/pdist-seuclidean-ml.txt

@@ -0,0 +1 @@
+   1.4330520e+01   1.4635426e+01   1.3450855e+01   1.4761140e+01   1.3508642e+01   1.5434417e+01   1.3887693e+01   1.5166776e+01   1.3966038e+01   1.4950451e+01   1.4564587e+01   1.3834201e+01   1.4347008e+01   1.5641962e+01   1.4689053e+01   1.4418720e+01   1.4545856e+01   1.4151822e+01   1.4669017e+01   1.5150750e+01   1.3770166e+01   1.3288969e+01   1.4048191e+01   1.4049959e+01   1.4164158e+01   1.3727834e+01   1.4074687e+01   1.4321303e+01   1.2497330e+01   1.3820273e+01   1.4441030e+01   1.4780222e+01   1.2504339e+01   1.5022245e+01   1.4263650e+01   1.3704507e+01   1.3694385e+01   1.3667517e+01   1.3177468e+01   1.4391931e+01   1.4893903e+01   1.4475753e+01   1.4440707e+01   1.3603096e+01   1.6889651e+01   1.4731174e+01   1.3337775e+01   1.5187532e+01   1.5667271e+01   1.4226037e+01   1.4203554e+01   1.5272898e+01   1.6031460e+01   1.5991549e+01   1.1855060e+01   1.4844776e+01   1.2475182e+01   1.4408126e+01   1.4836870e+01   1.3472986e+01   1.4089281e+01   1.1018298e+01   1.3183296e+01   1.4590802e+01   1.4404230e+01   1.2717623e+01   1.3983283e+01   1.4017133e+01   1.4608005e+01   1.4402553e+01   1.3977803e+01   1.4091040e+01   1.3977459e+01   1.2630449e+01   1.4160109e+01   1.3029417e+01   1.2654432e+01   1.2794946e+01   1.3194978e+01   1.4378745e+01   1.2431908e+01   1.3852651e+01   1.3748358e+01   1.4003568e+01   1.5066681e+01   1.5192826e+01   1.4370013e+01   1.5792545e+01   1.3547546e+01   1.4411543e+01   1.4794215e+01   1.4924312e+01   1.4789153e+01   1.4875055e+01   1.4208537e+01   1.2786148e+01   1.4882476e+01   1.3302010e+01   1.4354774e+01   1.4542129e+01   1.5889633e+01   1.2928185e+01   1.4877868e+01   1.2890902e+01   1.4406165e+01   1.4498123e+01   1.4303273e+01   1.3207002e+01   1.3954732e+01   1.4841248e+01   1.5427799e+01   1.4363463e+01   1.3976277e+01   1.4284878e+01   1.4457991e+01   1.3369469e+01   1.5246610e+01   1.4487573e+01   1.4525176e+01   1.4505865e+01   1.5037347e+01   1.3834927e+01   1.3758988e+01   1.3424987e+01   1.4914766e+01   1.3783923e+01   1.3434291e+01   1.2895927e+01   1.3870360e+01   1.3342977e+01   1.3094322e+01   1.3057847e+01   1.3322375e+01   1.4940650e+01   1.4476829e+01   1.4197503e+01   1.4597035e+01   1.2963234e+01   1.4011414e+01   1.3181409e+01   1.3339615e+01   1.3928735e+01   1.3508015e+01   1.3170749e+01   1.3529133e+01   1.3454724e+01   1.4883437e+01   1.4564565e+01   1.2474313e+01   1.4435790e+01   1.5285703e+01   1.3701736e+01   1.3578312e+01   1.4807311e+01   1.4281072e+01   1.2920213e+01   1.4427803e+01   1.1408611e+01   1.4097334e+01   1.2868115e+01   1.3903683e+01   1.3800332e+01   1.3439339e+01   1.4062651e+01   1.3242107e+01   1.4400424e+01   1.3826132e+01   1.5991146e+01   1.3118258e+01   1.5377390e+01   1.2858378e+01   1.5249567e+01   1.4081585e+01   1.4458052e+01   1.4175623e+01   1.4850069e+01   1.5506668e+01   1.5014770e+01   1.4337030e+01   1.5214705e+01   1.4803729e+01   1.3188675e+01   1.3437739e+01   1.3409394e+01   1.4607386e+01   1.5394271e+01   1.5946451e+01   1.3769364e+01   1.4181208e+01   1.2551765e+01

venv/Lib/site-packages/scipy/spatial/tests/data/pdist-spearman-ml.txt

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venv/Lib/site-packages/scipy/spatial/tests/data/random-uint-data.txt

@@ -0,0 +1,100 @@
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venv/Lib/site-packages/scipy/spatial/tests/data/selfdual-4d-polytope.txt

@@ -0,0 +1,27 @@
+# The facets of a self-dual 4-dim regular polytope
+# with 24 octahedron facets. Taken from cddlib.
+# Format b + Ax >= 0
+ 1  1  1  1  1
+ 1  1  1  1 -1
+ 1  1  1 -1  1
+ 1  1  1 -1 -1
+ 1  1 -1  1  1
+ 1  1 -1  1 -1
+ 1  1 -1 -1  1
+ 1  1 -1 -1 -1
+ 1 -1  1  1  1
+ 1 -1  1  1 -1
+ 1 -1  1 -1  1
+ 1 -1  1 -1 -1
+ 1 -1 -1  1  1
+ 1 -1 -1  1 -1
+ 1 -1 -1 -1  1
+ 1 -1 -1 -1 -1
+ 1  2  0  0  0
+ 1  0  2  0  0
+ 1  0  0  2  0
+ 1  0  0  0  2
+ 1 -2  0  0  0
+ 1  0 -2  0  0
+ 1  0  0 -2  0
+ 1  0  0  0 -2

venv/Lib/site-packages/scipy/spatial/tests/test__plotutils.py

@@ -0,0 +1,54 @@
+import pytest
+from numpy.testing import assert_, assert_array_equal, suppress_warnings
+try:
+    import matplotlib
+    matplotlib.rcParams['backend'] = 'Agg'
+    import matplotlib.pyplot as plt
+    has_matplotlib = True
+except Exception:
+    has_matplotlib = False
+
+from scipy.spatial import \
+     delaunay_plot_2d, voronoi_plot_2d, convex_hull_plot_2d, \
+     Delaunay, Voronoi, ConvexHull
+
+
+@pytest.mark.skipif(not has_matplotlib, reason="Matplotlib not available")
+class TestPlotting:
+    points = [(0,0), (0,1), (1,0), (1,1)]
+
+    def test_delaunay(self):
+        # Smoke test
+        fig = plt.figure()
+        obj = Delaunay(self.points)
+        s_before = obj.simplices.copy()
+        with suppress_warnings() as sup:
+            # filter can be removed when matplotlib 1.x is dropped
+            sup.filter(message="The ishold function was deprecated in version")
+            r = delaunay_plot_2d(obj, ax=fig.gca())
+        assert_array_equal(obj.simplices, s_before)  # shouldn't modify
+        assert_(r is fig)
+        delaunay_plot_2d(obj, ax=fig.gca())
+
+    def test_voronoi(self):
+        # Smoke test
+        fig = plt.figure()
+        obj = Voronoi(self.points)
+        with suppress_warnings() as sup:
+            # filter can be removed when matplotlib 1.x is dropped
+            sup.filter(message="The ishold function was deprecated in version")
+            r = voronoi_plot_2d(obj, ax=fig.gca())
+        assert_(r is fig)
+        voronoi_plot_2d(obj)
+        voronoi_plot_2d(obj, show_vertices=False)
+
+    def test_convex_hull(self):
+        # Smoke test
+        fig = plt.figure()
+        tri = ConvexHull(self.points)
+        with suppress_warnings() as sup:
+            # filter can be removed when matplotlib 1.x is dropped
+            sup.filter(message="The ishold function was deprecated in version")
+            r = convex_hull_plot_2d(tri, ax=fig.gca())
+        assert_(r is fig)
+        convex_hull_plot_2d(tri)

venv/Lib/site-packages/scipy/spatial/tests/test__procrustes.py

@@ -0,0 +1,116 @@
+import numpy as np
+from numpy.testing import assert_allclose, assert_equal, assert_almost_equal
+from pytest import raises as assert_raises
+
+from scipy.spatial import procrustes
+
+
+class TestProcrustes(object):
+    def setup_method(self):
+        """creates inputs"""
+        # an L
+        self.data1 = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], 'd')
+
+        # a larger, shifted, mirrored L
+        self.data2 = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], 'd')
+
+        # an L shifted up 1, right 1, and with point 4 shifted an extra .5
+        # to the right
+        # pointwise distance disparity with data1: 3*(2) + (1 + 1.5^2)
+        self.data3 = np.array([[2, 4], [2, 3], [2, 2], [3, 2.5]], 'd')
+
+        # data4, data5 are standardized (trace(A*A') = 1).
+        # procrustes should return an identical copy if they are used
+        # as the first matrix argument.
+        shiftangle = np.pi / 8
+        self.data4 = np.array([[1, 0], [0, 1], [-1, 0],
+                              [0, -1]], 'd') / np.sqrt(4)
+        self.data5 = np.array([[np.cos(shiftangle), np.sin(shiftangle)],
+                              [np.cos(np.pi / 2 - shiftangle),
+                               np.sin(np.pi / 2 - shiftangle)],
+                              [-np.cos(shiftangle),
+                               -np.sin(shiftangle)],
+                              [-np.cos(np.pi / 2 - shiftangle),
+                               -np.sin(np.pi / 2 - shiftangle)]],
+                              'd') / np.sqrt(4)
+
+    def test_procrustes(self):
+        # tests procrustes' ability to match two matrices.
+        #
+        # the second matrix is a rotated, shifted, scaled, and mirrored version
+        # of the first, in two dimensions only
+        #
+        # can shift, mirror, and scale an 'L'?
+        a, b, disparity = procrustes(self.data1, self.data2)
+        assert_allclose(b, a)
+        assert_almost_equal(disparity, 0.)
+
+        # if first mtx is standardized, leaves first mtx unchanged?
+        m4, m5, disp45 = procrustes(self.data4, self.data5)
+        assert_equal(m4, self.data4)
+
+        # at worst, data3 is an 'L' with one point off by .5
+        m1, m3, disp13 = procrustes(self.data1, self.data3)
+        #assert_(disp13 < 0.5 ** 2)
+
+    def test_procrustes2(self):
+        # procrustes disparity should not depend on order of matrices
+        m1, m3, disp13 = procrustes(self.data1, self.data3)
+        m3_2, m1_2, disp31 = procrustes(self.data3, self.data1)
+        assert_almost_equal(disp13, disp31)
+
+        # try with 3d, 8 pts per
+        rand1 = np.array([[2.61955202, 0.30522265, 0.55515826],
+                         [0.41124708, -0.03966978, -0.31854548],
+                         [0.91910318, 1.39451809, -0.15295084],
+                         [2.00452023, 0.50150048, 0.29485268],
+                         [0.09453595, 0.67528885, 0.03283872],
+                         [0.07015232, 2.18892599, -1.67266852],
+                         [0.65029688, 1.60551637, 0.80013549],
+                         [-0.6607528, 0.53644208, 0.17033891]])
+
+        rand3 = np.array([[0.0809969, 0.09731461, -0.173442],
+                         [-1.84888465, -0.92589646, -1.29335743],
+                         [0.67031855, -1.35957463, 0.41938621],
+                         [0.73967209, -0.20230757, 0.52418027],
+                         [0.17752796, 0.09065607, 0.29827466],
+                         [0.47999368, -0.88455717, -0.57547934],
+                         [-0.11486344, -0.12608506, -0.3395779],
+                         [-0.86106154, -0.28687488, 0.9644429]])
+        res1, res3, disp13 = procrustes(rand1, rand3)
+        res3_2, res1_2, disp31 = procrustes(rand3, rand1)
+        assert_almost_equal(disp13, disp31)
+
+    def test_procrustes_shape_mismatch(self):
+        assert_raises(ValueError, procrustes,
+                      np.array([[1, 2], [3, 4]]),
+                      np.array([[5, 6, 7], [8, 9, 10]]))
+
+    def test_procrustes_empty_rows_or_cols(self):
+        empty = np.array([[]])
+        assert_raises(ValueError, procrustes, empty, empty)
+
+    def test_procrustes_no_variation(self):
+        assert_raises(ValueError, procrustes,
+                      np.array([[42, 42], [42, 42]]),
+                      np.array([[45, 45], [45, 45]]))
+
+    def test_procrustes_bad_number_of_dimensions(self):
+        # fewer dimensions in one dataset
+        assert_raises(ValueError, procrustes,
+                      np.array([1, 1, 2, 3, 5, 8]),
+                      np.array([[1, 2], [3, 4]]))
+
+        # fewer dimensions in both datasets
+        assert_raises(ValueError, procrustes,
+                      np.array([1, 1, 2, 3, 5, 8]),
+                      np.array([1, 1, 2, 3, 5, 8]))
+
+        # zero dimensions
+        assert_raises(ValueError, procrustes, np.array(7), np.array(11))
+
+        # extra dimensions
+        assert_raises(ValueError, procrustes,
+                      np.array([[[11], [7]]]),
+                      np.array([[[5, 13]]]))
+

venv/Lib/site-packages/scipy/spatial/tests/test_hausdorff.py

@@ -0,0 +1,150 @@
+import numpy as np
+from numpy.testing import (assert_almost_equal,
+                           assert_array_equal,
+                           assert_equal,
+                           assert_)
+import pytest
+from scipy.spatial.distance import directed_hausdorff
+from scipy.spatial import distance
+from scipy._lib._util import check_random_state
+
+class TestHausdorff(object):
+    # Test various properties of the directed Hausdorff code.
+
+    def setup_method(self):
+        np.random.seed(1234)
+        random_angles = np.random.random(100) * np.pi * 2
+        random_columns = np.column_stack(
+            (random_angles, random_angles, np.zeros(100)))
+        random_columns[..., 0] = np.cos(random_columns[..., 0])
+        random_columns[..., 1] = np.sin(random_columns[..., 1])
+        random_columns_2 = np.column_stack(
+            (random_angles, random_angles, np.zeros(100)))
+        random_columns_2[1:, 0] = np.cos(random_columns_2[1:, 0]) * 2.0
+        random_columns_2[1:, 1] = np.sin(random_columns_2[1:, 1]) * 2.0
+        # move one point farther out so we don't have two perfect circles
+        random_columns_2[0, 0] = np.cos(random_columns_2[0, 0]) * 3.3
+        random_columns_2[0, 1] = np.sin(random_columns_2[0, 1]) * 3.3
+        self.path_1 = random_columns
+        self.path_2 = random_columns_2
+        self.path_1_4d = np.insert(self.path_1, 3, 5, axis=1)
+        self.path_2_4d = np.insert(self.path_2, 3, 27, axis=1)
+
+    def test_symmetry(self):
+        # Ensure that the directed (asymmetric) Hausdorff distance is
+        # actually asymmetric
+
+        forward = directed_hausdorff(self.path_1, self.path_2)[0]
+        reverse = directed_hausdorff(self.path_2, self.path_1)[0]
+        assert_(forward != reverse)
+
+    def test_brute_force_comparison_forward(self):
+        # Ensure that the algorithm for directed_hausdorff gives the
+        # same result as the simple / brute force approach in the
+        # forward direction.
+        actual = directed_hausdorff(self.path_1, self.path_2)[0]
+        # brute force over rows:
+        expected = max(np.amin(distance.cdist(self.path_1, self.path_2),
+                               axis=1))
+        assert_almost_equal(actual, expected, decimal=9)
+
+    def test_brute_force_comparison_reverse(self):
+        # Ensure that the algorithm for directed_hausdorff gives the
+        # same result as the simple / brute force approach in the
+        # reverse direction.
+        actual = directed_hausdorff(self.path_2, self.path_1)[0]
+        # brute force over columns:
+        expected = max(np.amin(distance.cdist(self.path_1, self.path_2), 
+                               axis=0))
+        assert_almost_equal(actual, expected, decimal=9)
+
+    def test_degenerate_case(self):
+        # The directed Hausdorff distance must be zero if both input
+        # data arrays match.
+        actual = directed_hausdorff(self.path_1, self.path_1)[0]
+        assert_almost_equal(actual, 0.0, decimal=9)
+
+    def test_2d_data_forward(self):
+        # Ensure that 2D data is handled properly for a simple case
+        # relative to brute force approach.
+        actual = directed_hausdorff(self.path_1[..., :2],
+                                    self.path_2[..., :2])[0]
+        expected = max(np.amin(distance.cdist(self.path_1[..., :2],
+                                              self.path_2[..., :2]),
+                               axis=1))
+        assert_almost_equal(actual, expected, decimal=9)
+
+    def test_4d_data_reverse(self):
+        # Ensure that 4D data is handled properly for a simple case
+        # relative to brute force approach.
+        actual = directed_hausdorff(self.path_2_4d, self.path_1_4d)[0]
+        # brute force over columns:
+        expected = max(np.amin(distance.cdist(self.path_1_4d, self.path_2_4d), 
+                               axis=0))
+        assert_almost_equal(actual, expected, decimal=9)
+
+    def test_indices(self):
+        # Ensure that correct point indices are returned -- they should
+        # correspond to the Hausdorff pair
+        path_simple_1 = np.array([[-1,-12],[0,0], [1,1], [3,7], [1,2]])
+        path_simple_2 = np.array([[0,0], [1,1], [4,100], [10,9]])
+        actual = directed_hausdorff(path_simple_2, path_simple_1)[1:]
+        expected = (2, 3)
+        assert_array_equal(actual, expected)
+
+    def test_random_state(self):
+        # ensure that the global random state is not modified because
+        # the directed Hausdorff algorithm uses randomization
+        rs = check_random_state(None)
+        old_global_state = rs.get_state()
+        directed_hausdorff(self.path_1, self.path_2)
+        rs2 = check_random_state(None)
+        new_global_state = rs2.get_state()
+        assert_equal(new_global_state, old_global_state)
+
+    def test_random_state_None_int(self):
+        # check that seed values of None or int do not alter global
+        # random state
+        for seed in [None, 27870671]:
+            rs = check_random_state(None)
+            old_global_state = rs.get_state()
+            directed_hausdorff(self.path_1, self.path_2, seed)
+            rs2 = check_random_state(None)
+            new_global_state = rs2.get_state()
+            assert_equal(new_global_state, old_global_state)
+
+    def test_invalid_dimensions(self):
+        # Ensure that a ValueError is raised when the number of columns
+        # is not the same
+        np.random.seed(1234)
+        A = np.random.rand(3, 2)
+        B = np.random.rand(4, 5)
+        with pytest.raises(ValueError):
+            directed_hausdorff(A, B)
+
+    @pytest.mark.parametrize("A, B, seed, expected", [
+        # the two cases from gh-11332
+        ([(0,0)],
+         [(0,1), (0,0)],
+         0,
+         (0.0, 0, 1)),
+        ([(0,0)],
+         [(0,1), (0,0)],
+         1,
+         (0.0, 0, 1)),
+        # slightly more complex case
+        ([(-5, 3), (0,0)],
+         [(0,1), (0,0), (-5, 3)],
+         77098,
+         # the maximum minimum distance will
+         # be the last one found, but a unique
+         # solution is not guaranteed more broadly
+         (0.0, 1, 1)),
+    ])
+    def test_subsets(self, A, B, seed, expected):
+        # verify fix for gh-11332
+        actual = directed_hausdorff(u=A, v=B, seed=seed)
+        # check distance
+        assert_almost_equal(actual[0], expected[0], decimal=9)
+        # check indices
+        assert actual[1:] == expected[1:]

venv/Lib/site-packages/scipy/spatial/tests/test_slerp.py

@@ -0,0 +1,383 @@
+from __future__ import division, absolute_import, print_function
+
+import numpy as np
+from numpy.testing import assert_allclose
+
+import pytest
+from scipy.spatial import geometric_slerp
+
+
+def _generate_spherical_points(ndim=3, n_pts=2):
+    # generate uniform points on sphere
+    # see: https://stackoverflow.com/a/23785326
+    # tentatively extended to arbitrary dims
+    # for 0-sphere it will always produce antipodes
+    np.random.seed(123)
+    points = np.random.normal(size=(n_pts, ndim))
+    points /= np.linalg.norm(points, axis=1)[:, np.newaxis]
+    return points[0], points[1]
+
+
+class TestGeometricSlerp(object):
+    # Test various properties of the geometric slerp code
+
+    @pytest.mark.parametrize("n_dims", [2, 3, 5, 7, 9])
+    @pytest.mark.parametrize("n_pts", [0, 3, 17])
+    def test_shape_property(self, n_dims, n_pts):
+        # geometric_slerp output shape should match
+        # input dimensionality & requested number
+        # of interpolation points
+        start, end = _generate_spherical_points(n_dims, 2)
+
+        actual = geometric_slerp(start=start,
+                                 end=end,
+                                 t=np.linspace(0, 1, n_pts))
+
+        assert actual.shape == (n_pts, n_dims)
+
+    @pytest.mark.parametrize("n_dims", [2, 3, 5, 7, 9])
+    @pytest.mark.parametrize("n_pts", [3, 17])
+    def test_include_ends(self, n_dims, n_pts):
+        # geometric_slerp should return a data structure
+        # that includes the start and end coordinates
+        # when t includes 0 and 1 ends
+        # this is convenient for plotting surfaces represented
+        # by interpolations for example
+
+        # the generator doesn't work so well for the unit
+        # sphere (it always produces antipodes), so use
+        # custom values there
+        start, end = _generate_spherical_points(n_dims, 2)
+
+        actual = geometric_slerp(start=start,
+                                 end=end,
+                                 t=np.linspace(0, 1, n_pts))
+
+        assert_allclose(actual[0], start)
+        assert_allclose(actual[-1], end)
+
+    @pytest.mark.parametrize("start, end", [
+        # both arrays are not flat
+        (np.zeros((1, 3)), np.ones((1, 3))),
+        # only start array is not flat
+        (np.zeros((1, 3)), np.ones(3)),
+        # only end array is not flat
+        (np.zeros(1), np.ones((3, 1))),
+        ])
+    def test_input_shape_flat(self, start, end):
+        # geometric_slerp should handle input arrays that are
+        # not flat appropriately
+        with pytest.raises(ValueError, match='one-dimensional'):
+            geometric_slerp(start=start,
+                            end=end,
+                            t=np.linspace(0, 1, 10))
+
+    @pytest.mark.parametrize("start, end", [
+        # 7-D and 3-D ends
+        (np.zeros(7), np.ones(3)),
+        # 2-D and 1-D ends
+        (np.zeros(2), np.ones(1)),
+        # empty, "3D" will also get caught this way
+        (np.array([]), np.ones(3)),
+        ])
+    def test_input_dim_mismatch(self, start, end):
+        # geometric_slerp must appropriately handle cases where
+        # an interpolation is attempted across two different
+        # dimensionalities
+        with pytest.raises(ValueError, match='dimensions'):
+            geometric_slerp(start=start,
+                            end=end,
+                            t=np.linspace(0, 1, 10))
+
+    @pytest.mark.parametrize("start, end", [
+        # both empty
+        (np.array([]), np.array([])),
+        ])
+    def test_input_at_least1d(self, start, end):
+        # empty inputs to geometric_slerp must
+        # be handled appropriately when not detected
+        # by mismatch
+        with pytest.raises(ValueError, match='at least two-dim'):
+            geometric_slerp(start=start,
+                            end=end,
+                            t=np.linspace(0, 1, 10))
+
+    @pytest.mark.parametrize("start, end, expected", [
+        # North and South Poles are definitely antipodes
+        # but should be handled gracefully now
+        (np.array([0, 0, 1.0]), np.array([0, 0, -1.0]), "warning"),
+        # this case will issue a warning & be handled
+        # gracefully as well;
+        # North Pole was rotated very slightly
+        # using r = R.from_euler('x', 0.035, degrees=True)
+        # to achieve Euclidean distance offset from diameter by
+        # 9.328908379124812e-08, within the default tol
+        (np.array([0.00000000e+00,
+                  -6.10865200e-04,
+                  9.99999813e-01]), np.array([0, 0, -1.0]), "warning"),
+        # this case should succeed without warning because a
+        # sufficiently large
+        # rotation was applied to North Pole point to shift it
+        # to a Euclidean distance of 2.3036691931821451e-07
+        # from South Pole, which is larger than tol
+        (np.array([0.00000000e+00,
+                  -9.59930941e-04,
+                  9.99999539e-01]), np.array([0, 0, -1.0]), "success"),
+        ])
+    def test_handle_antipodes(self, start, end, expected):
+        # antipodal points must be handled appropriately;
+        # there are an infinite number of possible geodesic
+        # interpolations between them in higher dims
+        if expected == "warning":
+            with pytest.warns(UserWarning, match='antipodes'):
+                res = geometric_slerp(start=start,
+                                      end=end,
+                                      t=np.linspace(0, 1, 10))
+        else:
+            res = geometric_slerp(start=start,
+                                  end=end,
+                                  t=np.linspace(0, 1, 10))
+
+        # antipodes or near-antipodes should still produce
+        # slerp paths on the surface of the sphere (but they
+        # may be ambiguous):
+        assert_allclose(np.linalg.norm(res, axis=1), 1.0)
+
+    @pytest.mark.parametrize("start, end, expected", [
+        # 2-D with n_pts=4 (two new interpolation points)
+        # this is an actual circle
+        (np.array([1, 0]),
+         np.array([0, 1]),
+         np.array([[1, 0],
+                   [np.sqrt(3) / 2, 0.5],  # 30 deg on unit circle
+                   [0.5, np.sqrt(3) / 2],  # 60 deg on unit circle
+                   [0, 1]])),
+        # likewise for 3-D (add z = 0 plane)
+        # this is an ordinary sphere
+        (np.array([1, 0, 0]),
+         np.array([0, 1, 0]),
+         np.array([[1, 0, 0],
+                   [np.sqrt(3) / 2, 0.5, 0],
+                   [0.5, np.sqrt(3) / 2, 0],
+                   [0, 1, 0]])),
+        # for 5-D, pad more columns with constants
+        # zeros are easiest--non-zero values on unit
+        # circle are more difficult to reason about
+        # at higher dims
+        (np.array([1, 0, 0, 0, 0]),
+         np.array([0, 1, 0, 0, 0]),
+         np.array([[1, 0, 0, 0, 0],
+                   [np.sqrt(3) / 2, 0.5, 0, 0, 0],
+                   [0.5, np.sqrt(3) / 2, 0, 0, 0],
+                   [0, 1, 0, 0, 0]])),
+
+    ])
+    def test_straightforward_examples(self, start, end, expected):
+        # some straightforward interpolation tests, sufficiently
+        # simple to use the unit circle to deduce expected values;
+        # for larger dimensions, pad with constants so that the
+        # data is N-D but simpler to reason about
+        actual = geometric_slerp(start=start,
+                                 end=end,
+                                 t=np.linspace(0, 1, 4))
+        assert_allclose(actual, expected, atol=1e-16)
+
+    @pytest.mark.parametrize("t", [
+        # both interval ends clearly violate limits
+        np.linspace(-20, 20, 300),
+        # only one interval end violating limit slightly
+        np.linspace(-0.0001, 0.0001, 17),
+        ])
+    def test_t_values_limits(self, t):
+        # geometric_slerp() should appropriately handle
+        # interpolation parameters < 0 and > 1
+        with pytest.raises(ValueError, match='interpolation parameter'):
+            _ = geometric_slerp(start=np.array([1, 0]),
+                                end=np.array([0, 1]),
+                                t=t)
+
+    @pytest.mark.parametrize("start, end", [
+        (np.array([1]),
+         np.array([0])),
+        (np.array([0]),
+         np.array([1])),
+        (np.array([-17.7]),
+         np.array([165.9])),
+     ])
+    def test_0_sphere_handling(self, start, end):
+        # it does not make sense to interpolate the set of
+        # two points that is the 0-sphere
+        with pytest.raises(ValueError, match='at least two-dim'):
+            _ = geometric_slerp(start=start,
+                                end=end,
+                                t=np.linspace(0, 1, 4))
+
+    @pytest.mark.parametrize("tol", [
+        # an integer currently raises
+        5,
+        # string raises
+        "7",
+        # list and arrays also raise
+        [5, 6, 7], np.array(9.0),
+        ])
+    def test_tol_type(self, tol):
+        # geometric_slerp() should raise if tol is not
+        # a suitable float type
+        with pytest.raises(ValueError, match='must be a float'):
+            _ = geometric_slerp(start=np.array([1, 0]),
+                                end=np.array([0, 1]),
+                                t=np.linspace(0, 1, 5),
+                                tol=tol)
+
+    @pytest.mark.parametrize("tol", [
+        -5e-6,
+        -7e-10,
+        ])
+    def test_tol_sign(self, tol):
+        # geometric_slerp() currently handles negative
+        # tol values, as long as they are floats
+        _ = geometric_slerp(start=np.array([1, 0]),
+                            end=np.array([0, 1]),
+                            t=np.linspace(0, 1, 5),
+                            tol=tol)
+
+    @pytest.mark.parametrize("start, end", [
+        # 1-sphere (circle) with one point at origin
+        # and the other on the circle
+        (np.array([1, 0]), np.array([0, 0])),
+        # 2-sphere (normal sphere) with both points
+        # just slightly off sphere by the same amount
+        # in different directions
+        (np.array([1 + 1e-6, 0, 0]),
+         np.array([0, 1 - 1e-6, 0])),
+        # same thing in 4-D
+        (np.array([1 + 1e-6, 0, 0, 0]),
+         np.array([0, 1 - 1e-6, 0, 0])),
+        ])
+    def test_unit_sphere_enforcement(self, start, end):
+        # geometric_slerp() should raise on input that clearly
+        # cannot be on an n-sphere of radius 1
+        with pytest.raises(ValueError, match='unit n-sphere'):
+            geometric_slerp(start=start,
+                            end=end,
+                            t=np.linspace(0, 1, 5))
+
+    @pytest.mark.parametrize("start, end", [
+        # 1-sphere 45 degree case
+        (np.array([1, 0]),
+         np.array([np.sqrt(2) / 2.,
+                   np.sqrt(2) / 2.])),
+        # 2-sphere 135 degree case
+        (np.array([1, 0]),
+         np.array([-np.sqrt(2) / 2.,
+                   np.sqrt(2) / 2.])),
+        ])
+    @pytest.mark.parametrize("t_func", [
+        np.linspace, np.logspace])
+    def test_order_handling(self, start, end, t_func):
+        # geometric_slerp() should handle scenarios with
+        # ascending and descending t value arrays gracefully;
+        # results should simply be reversed
+
+        # for scrambled / unsorted parameters, the same values
+        # should be returned, just in scrambled order
+
+        num_t_vals = 20
+        np.random.seed(789)
+        forward_t_vals = t_func(0, 10, num_t_vals)
+        # normalize to max of 1
+        forward_t_vals /= forward_t_vals.max()
+        reverse_t_vals = np.flipud(forward_t_vals)
+        shuffled_indices = np.arange(num_t_vals)
+        np.random.shuffle(shuffled_indices)
+        scramble_t_vals = forward_t_vals.copy()[shuffled_indices]
+
+        forward_results = geometric_slerp(start=start,
+                                          end=end,
+                                          t=forward_t_vals)
+        reverse_results = geometric_slerp(start=start,
+                                          end=end,
+                                          t=reverse_t_vals)
+        scrambled_results = geometric_slerp(start=start,
+                                            end=end,
+                                            t=scramble_t_vals)
+
+        # check fidelity to input order
+        assert_allclose(forward_results, np.flipud(reverse_results))
+        assert_allclose(forward_results[shuffled_indices],
+                        scrambled_results)
+
+    @pytest.mark.parametrize("t", [
+        # string:
+        "15, 5, 7",
+        # complex numbers currently produce a warning
+        # but not sure we need to worry about it too much:
+        # [3 + 1j, 5 + 2j],
+        ])
+    def test_t_values_conversion(self, t):
+        with pytest.raises(ValueError):
+            _ = geometric_slerp(start=np.array([1]),
+                                end=np.array([0]),
+                                t=t)
+
+    def test_accept_arraylike(self):
+        # array-like support requested by reviewer
+        # in gh-10380
+        actual = geometric_slerp([1, 0], [0, 1], [0, 1/3, 0.5, 2/3, 1])
+
+        # expected values are based on visual inspection
+        # of the unit circle for the progressions along
+        # the circumference provided in t
+        expected = np.array([[1, 0],
+                             [np.sqrt(3) / 2, 0.5],
+                             [np.sqrt(2) / 2,
+                              np.sqrt(2) / 2],
+                             [0.5, np.sqrt(3) / 2],
+                             [0, 1]], dtype=np.float64)
+        # Tyler's original Cython implementation of geometric_slerp
+        # can pass at atol=0 here, but on balance we will accept
+        # 1e-16 for an implementation that avoids Cython and
+        # makes up accuracy ground elsewhere
+        assert_allclose(actual, expected, atol=1e-16)
+
+    def test_scalar_t(self):
+        # when t is a scalar, return value is a single
+        # interpolated point of the appropriate dimensionality
+        # requested by reviewer in gh-10380
+        actual = geometric_slerp([1, 0], [0, 1], 0.5)
+        expected = np.array([np.sqrt(2) / 2,
+                             np.sqrt(2) / 2], dtype=np.float64)
+        assert actual.shape == (2,)
+        assert_allclose(actual, expected)
+
+    @pytest.mark.parametrize('start', [
+        np.array([1, 0, 0]),
+        np.array([0, 1]),
+        ])
+    def test_degenerate_input(self, start):
+        # handle start == end with repeated value
+        # like np.linspace
+        expected = [start] * 5
+        actual = geometric_slerp(start=start,
+                                 end=start,
+                                 t=np.linspace(0, 1, 5))
+        assert_allclose(actual, expected)
+
+    @pytest.mark.parametrize('k', np.logspace(-10, -1, 10))
+    def test_numerical_stability_pi(self, k):
+        # geometric_slerp should have excellent numerical
+        # stability for angles approaching pi between
+        # the start and end points
+        angle = np.pi - k
+        ts = np.linspace(0, 1, 100)
+        P = np.array([1, 0, 0, 0])
+        Q = np.array([np.cos(angle), np.sin(angle), 0, 0])
+        # the test should only be enforced for cases where
+        # geometric_slerp determines that the input is actually
+        # on the unit sphere
+        with np.testing.suppress_warnings() as sup:
+            sup.filter(UserWarning)
+            result = geometric_slerp(P, Q, ts, 1e-18)
+            norms = np.linalg.norm(result, axis=1)
+            error = np.max(np.abs(norms - 1))
+            assert error < 4e-15

venv/Lib/site-packages/scipy/spatial/tests/test_spherical_voronoi.py

@@ -0,0 +1,365 @@
+import numpy as np
+import itertools
+from numpy.testing import (assert_equal,
+                           assert_almost_equal,
+                           assert_array_equal,
+                           assert_array_almost_equal,
+                           suppress_warnings)
+import pytest
+from pytest import raises as assert_raises
+from pytest import warns as assert_warns
+from scipy.spatial import SphericalVoronoi, distance
+from scipy.optimize import linear_sum_assignment
+from scipy.constants import golden as phi
+from scipy.special import gamma
+
+
+TOL = 1E-10
+
+
+def _generate_tetrahedron():
+    return np.array([[1, 1, 1], [1, -1, -1], [-1, 1, -1], [-1, -1, 1]])
+
+
+def _generate_cube():
+    return np.array(list(itertools.product([-1, 1.], repeat=3)))
+
+
+def _generate_octahedron():
+    return np.array([[-1, 0, 0], [+1, 0, 0], [0, -1, 0],
+                     [0, +1, 0], [0, 0, -1], [0, 0, +1]])
+
+
+def _generate_dodecahedron():
+
+    x1 = _generate_cube()
+    x2 = np.array([[0, -phi, -1 / phi],
+                   [0, -phi, +1 / phi],
+                   [0, +phi, -1 / phi],
+                   [0, +phi, +1 / phi]])
+    x3 = np.array([[-1 / phi, 0, -phi],
+                   [+1 / phi, 0, -phi],
+                   [-1 / phi, 0, +phi],
+                   [+1 / phi, 0, +phi]])
+    x4 = np.array([[-phi, -1 / phi, 0],
+                   [-phi, +1 / phi, 0],
+                   [+phi, -1 / phi, 0],
+                   [+phi, +1 / phi, 0]])
+    return np.concatenate((x1, x2, x3, x4))
+
+
+def _generate_icosahedron():
+    x = np.array([[0, -1, -phi],
+                  [0, -1, +phi],
+                  [0, +1, -phi],
+                  [0, +1, +phi]])
+    return np.concatenate([np.roll(x, i, axis=1) for i in range(3)])
+
+
+def _generate_polytope(name):
+    polygons = ["triangle", "square", "pentagon", "hexagon", "heptagon",
+                "octagon", "nonagon", "decagon", "undecagon", "dodecagon"]
+    polyhedra = ["tetrahedron", "cube", "octahedron", "dodecahedron",
+                 "icosahedron"]
+    if name not in polygons and name not in polyhedra:
+        raise ValueError("unrecognized polytope")
+
+    if name in polygons:
+        n = polygons.index(name) + 3
+        thetas = np.linspace(0, 2 * np.pi, n, endpoint=False)
+        p = np.vstack([np.cos(thetas), np.sin(thetas)]).T
+    elif name == "tetrahedron":
+        p = _generate_tetrahedron()
+    elif name == "cube":
+        p = _generate_cube()
+    elif name == "octahedron":
+        p = _generate_octahedron()
+    elif name == "dodecahedron":
+        p = _generate_dodecahedron()
+    elif name == "icosahedron":
+        p = _generate_icosahedron()
+
+    return p / np.linalg.norm(p, axis=1, keepdims=True)
+
+
+def _hypersphere_area(dim, radius):
+    # https://en.wikipedia.org/wiki/N-sphere#Closed_forms
+    return 2 * np.pi**(dim / 2) / gamma(dim / 2) * radius**(dim - 1)
+
+
+def _sample_sphere(n, dim, seed=None):
+    # Sample points uniformly at random from the hypersphere
+    rng = np.random.RandomState(seed=seed)
+    points = rng.randn(n, dim)
+    points /= np.linalg.norm(points, axis=1, keepdims=True)
+    return points
+
+
+class TestSphericalVoronoi(object):
+
+    def setup_method(self):
+        self.points = np.array([
+            [-0.78928481, -0.16341094, 0.59188373],
+            [-0.66839141, 0.73309634, 0.12578818],
+            [0.32535778, -0.92476944, -0.19734181],
+            [-0.90177102, -0.03785291, -0.43055335],
+            [0.71781344, 0.68428936, 0.12842096],
+            [-0.96064876, 0.23492353, -0.14820556],
+            [0.73181537, -0.22025898, -0.6449281],
+            [0.79979205, 0.54555747, 0.25039913]]
+        )
+
+        # Issue #9386
+        self.hemisphere_points = np.array([
+            [0.88610999, -0.42383021, 0.18755541],
+            [0.51980039, -0.72622668, 0.4498915],
+            [0.56540011, -0.81629197, -0.11827989],
+            [0.69659682, -0.69972598, 0.15854467]])
+
+        # Issue #8859
+        phi = np.linspace(0, 2 * np.pi, 10, endpoint=False)    # azimuth angle
+        theta = np.linspace(0.001, np.pi * 0.4, 5)    # polar angle
+        theta = theta[np.newaxis, :].T
+
+        phiv, thetav = np.meshgrid(phi, theta)
+        phiv = np.reshape(phiv, (50, 1))
+        thetav = np.reshape(thetav, (50, 1))
+
+        x = np.cos(phiv) * np.sin(thetav)
+        y = np.sin(phiv) * np.sin(thetav)
+        z = np.cos(thetav)
+        self.hemisphere_points2 = np.concatenate([x, y, z], axis=1)
+
+    def test_constructor(self):
+        center = np.array([1, 2, 3])
+        radius = 2
+        s1 = SphericalVoronoi(self.points)
+        # user input checks in SphericalVoronoi now require
+        # the radius / center to match the generators so adjust
+        # accordingly here
+        s2 = SphericalVoronoi(self.points * radius, radius)
+        s3 = SphericalVoronoi(self.points + center, center=center)
+        s4 = SphericalVoronoi(self.points * radius + center, radius, center)
+        assert_array_equal(s1.center, np.array([0, 0, 0]))
+        assert_equal(s1.radius, 1)
+        assert_array_equal(s2.center, np.array([0, 0, 0]))
+        assert_equal(s2.radius, 2)
+        assert_array_equal(s3.center, center)
+        assert_equal(s3.radius, 1)
+        assert_array_equal(s4.center, center)
+        assert_equal(s4.radius, radius)
+
+    def test_vertices_regions_translation_invariance(self):
+        sv_origin = SphericalVoronoi(self.points)
+        center = np.array([1, 1, 1])
+        sv_translated = SphericalVoronoi(self.points + center, center=center)
+        assert_equal(sv_origin.regions, sv_translated.regions)
+        assert_array_almost_equal(sv_origin.vertices + center,
+                                  sv_translated.vertices)
+
+    def test_vertices_regions_scaling_invariance(self):
+        sv_unit = SphericalVoronoi(self.points)
+        sv_scaled = SphericalVoronoi(self.points * 2, 2)
+        assert_equal(sv_unit.regions, sv_scaled.regions)
+        assert_array_almost_equal(sv_unit.vertices * 2,
+                                  sv_scaled.vertices)
+
+    def test_old_radius_api(self):
+        sv_unit = SphericalVoronoi(self.points, radius=1)
+        with suppress_warnings() as sup:
+            sup.filter(DeprecationWarning, "`radius` is `None`")
+            sv = SphericalVoronoi(self.points, None)
+            assert_array_almost_equal(sv_unit.vertices, sv.vertices)
+
+    def test_old_radius_api_warning(self):
+        with assert_warns(DeprecationWarning):
+            SphericalVoronoi(self.points, None)
+
+    def test_sort_vertices_of_regions(self):
+        sv = SphericalVoronoi(self.points)
+        unsorted_regions = sv.regions
+        sv.sort_vertices_of_regions()
+        assert_equal(sorted(sv.regions), sorted(unsorted_regions))
+
+    def test_sort_vertices_of_regions_flattened(self):
+        expected = sorted([[0, 6, 5, 2, 3], [2, 3, 10, 11, 8, 7], [0, 6, 4, 1],
+                           [4, 8, 7, 5, 6], [9, 11, 10], [2, 7, 5],
+                           [1, 4, 8, 11, 9], [0, 3, 10, 9, 1]])
+        expected = list(itertools.chain(*sorted(expected)))  # type: ignore
+        sv = SphericalVoronoi(self.points)
+        sv.sort_vertices_of_regions()
+        actual = list(itertools.chain(*sorted(sv.regions)))
+        assert_array_equal(actual, expected)
+
+    def test_sort_vertices_of_regions_dimensionality(self):
+        points = np.array([[1, 0, 0, 0],
+                           [0, 1, 0, 0],
+                           [0, 0, 1, 0],
+                           [0, 0, 0, 1],
+                           [0.5, 0.5, 0.5, 0.5]])
+        with pytest.raises(TypeError, match="three-dimensional"):
+            sv = SphericalVoronoi(points)
+            sv.sort_vertices_of_regions()
+
+    def test_num_vertices(self):
+        # for any n >= 3, a spherical Voronoi diagram has 2n - 4
+        # vertices; this is a direct consequence of Euler's formula
+        # as explained by Dinis and Mamede (2010) Proceedings of the
+        # 2010 International Symposium on Voronoi Diagrams in Science
+        # and Engineering
+        sv = SphericalVoronoi(self.points)
+        expected = self.points.shape[0] * 2 - 4
+        actual = sv.vertices.shape[0]
+        assert_equal(actual, expected)
+
+    def test_voronoi_circles(self):
+        sv = SphericalVoronoi(self.points)
+        for vertex in sv.vertices:
+            distances = distance.cdist(sv.points, np.array([vertex]))
+            closest = np.array(sorted(distances)[0:3])
+            assert_almost_equal(closest[0], closest[1], 7, str(vertex))
+            assert_almost_equal(closest[0], closest[2], 7, str(vertex))
+
+    def test_duplicate_point_handling(self):
+        # an exception should be raised for degenerate generators
+        # related to Issue# 7046
+        self.degenerate = np.concatenate((self.points, self.points))
+        with assert_raises(ValueError):
+            SphericalVoronoi(self.degenerate)
+
+    def test_incorrect_radius_handling(self):
+        # an exception should be raised if the radius provided
+        # cannot possibly match the input generators
+        with assert_raises(ValueError):
+            SphericalVoronoi(self.points, radius=0.98)
+
+    def test_incorrect_center_handling(self):
+        # an exception should be raised if the center provided
+        # cannot possibly match the input generators
+        with assert_raises(ValueError):
+            SphericalVoronoi(self.points, center=[0.1, 0, 0])
+
+    def test_single_hemisphere_handling(self):
+        # Test solution of Issues #9386, #8859
+
+        for points in [self.hemisphere_points, self.hemisphere_points2]:
+            sv = SphericalVoronoi(points)
+            triangles = sv.points[sv._simplices]
+            dots = np.einsum('ij,ij->i', sv.vertices, triangles[:, 0])
+            circumradii = np.arccos(np.clip(dots, -1, 1))
+            assert np.max(circumradii) > np.pi / 2
+
+    @pytest.mark.parametrize("n", [1, 2, 10])
+    @pytest.mark.parametrize("center", [(0, 0, 0), (1, 2, 3)])
+    def test_rank_deficient(self, n, center):
+        thetas = np.linspace(0, 2 * np.pi, n, endpoint=False)
+        points = np.vstack([np.sin(thetas), np.cos(thetas), np.zeros(n)]).T
+        with pytest.raises(ValueError, match="Rank of input points"):
+            SphericalVoronoi(points + center, center=center)
+
+    @pytest.mark.parametrize("dim", range(2, 7))
+    def test_higher_dimensions(self, dim):
+        n = 100
+        points = _sample_sphere(n, dim, seed=0)
+        sv = SphericalVoronoi(points)
+        assert sv.vertices.shape[1] == dim
+        assert len(sv.regions) == n
+
+        # verify Euler characteristic
+        cell_counts = []
+        simplices = np.sort(sv._simplices)
+        for i in range(1, dim + 1):
+            cells = []
+            for indices in itertools.combinations(range(dim), i):
+                cells.append(simplices[:, list(indices)])
+            cells = np.unique(np.concatenate(cells), axis=0)
+            cell_counts.append(len(cells))
+        expected_euler = 1 + (-1)**(dim-1)
+        actual_euler = sum([(-1)**i * e for i, e in enumerate(cell_counts)])
+        assert expected_euler == actual_euler
+
+    @pytest.mark.parametrize("dim", range(2, 7))
+    def test_cross_polytope_regions(self, dim):
+        # The hypercube is the dual of the cross-polytope, so the voronoi
+        # vertices of the cross-polytope lie on the points of the hypercube.
+
+        # generate points of the cross-polytope
+        points = np.concatenate((-np.eye(dim), np.eye(dim)))
+        sv = SphericalVoronoi(points)
+        assert all([len(e) == 2**(dim - 1) for e in sv.regions])
+
+        # generate points of the hypercube
+        expected = np.vstack(list(itertools.product([-1, 1], repeat=dim)))
+        expected = expected.astype(np.float64) / np.sqrt(dim)
+
+        # test that Voronoi vertices are correctly placed
+        dist = distance.cdist(sv.vertices, expected)
+        res = linear_sum_assignment(dist)
+        assert dist[res].sum() < TOL
+
+    @pytest.mark.parametrize("dim", range(2, 4))
+    def test_hypercube_regions(self, dim):
+        # The cross-polytope is the dual of the hypercube, so the voronoi
+        # vertices of the hypercube lie on the points of the cross-polytope.
+
+        # generate points of the hypercube
+        points = np.vstack(list(itertools.product([-1, 1], repeat=dim)))
+        points = points.astype(np.float64) / np.sqrt(dim)
+        sv = SphericalVoronoi(points)
+
+        # generate points of the cross-polytope
+        expected = np.concatenate((-np.eye(dim), np.eye(dim)))
+
+        # test that Voronoi vertices are correctly placed
+        dist = distance.cdist(sv.vertices, expected)
+        res = linear_sum_assignment(dist)
+        assert dist[res].sum() < TOL
+
+    @pytest.mark.parametrize("n", [10, 500])
+    @pytest.mark.parametrize("dim", [2, 3])
+    @pytest.mark.parametrize("radius", [0.5, 1, 2])
+    @pytest.mark.parametrize("shift", [False, True])
+    @pytest.mark.parametrize("single_hemisphere", [False, True])
+    def test_area_reconstitution(self, n, dim, radius, shift,
+                                 single_hemisphere):
+        points = _sample_sphere(n, dim, seed=0)
+
+        # move all points to one side of the sphere for single-hemisphere test
+        if single_hemisphere:
+            points[:, 0] = np.abs(points[:, 0])
+
+        center = (np.arange(dim) + 1) * shift
+        points = radius * points + center
+
+        sv = SphericalVoronoi(points, radius=radius, center=center)
+        areas = sv.calculate_areas()
+        assert_almost_equal(areas.sum(), _hypersphere_area(dim, radius))
+
+    @pytest.mark.parametrize("poly", ["triangle", "dodecagon",
+                                      "tetrahedron", "cube", "octahedron",
+                                      "dodecahedron", "icosahedron"])
+    def test_equal_area_reconstitution(self, poly):
+        points = _generate_polytope(poly)
+        n, dim = points.shape
+        sv = SphericalVoronoi(points)
+        areas = sv.calculate_areas()
+        assert_almost_equal(areas, _hypersphere_area(dim, 1) / n)
+
+    def test_area_unsupported_dimension(self):
+        dim = 4
+        points = np.concatenate((-np.eye(dim), np.eye(dim)))
+        sv = SphericalVoronoi(points)
+        with pytest.raises(TypeError, match="Only supported"):
+            sv.calculate_areas()
+
+    @pytest.mark.parametrize("radius", [1, 1.])
+    @pytest.mark.parametrize("center", [None, (1, 2, 3), (1., 2., 3.)])
+    def test_attribute_types(self, radius, center):
+        points = radius * self.points
+        if center is not None:
+            points += center
+
+        sv = SphericalVoronoi(points, radius=radius, center=center)
+        assert sv.points.dtype is np.dtype(np.float64)
+        assert sv.center.dtype is np.dtype(np.float64)
+        assert isinstance(sv.radius, float)

venv/Lib/site-packages/scipy/spatial/transform/__init__.py

@@ -0,0 +1,26 @@
+"""
+Spatial Transformations (:mod:`scipy.spatial.transform`)
+========================================================
+
+.. currentmodule:: scipy.spatial.transform
+
+This package implements various spatial transformations. For now,
+only rotations are supported.
+
+Rotations in 3 dimensions
+-------------------------
+.. autosummary::
+   :toctree: generated/
+
+   Rotation
+   Slerp
+   RotationSpline
+"""
+from .rotation import Rotation, Slerp
+from ._rotation_spline import RotationSpline
+
+__all__ = ['Rotation', 'Slerp', 'RotationSpline']
+
+from scipy._lib._testutils import PytestTester
+test = PytestTester(__name__)
+del PytestTester

venv/Lib/site-packages/scipy/spatial/transform/_rotation_groups.py

@@ -0,0 +1,140 @@
+import numpy as np
+from scipy.constants import golden as phi
+
+
+def icosahedral(cls):
+    g1 = tetrahedral(cls).as_quat()
+    a = 0.5
+    b = 0.5 / phi
+    c = phi / 2
+    g2 = np.array([[+a, +b, +c, 0],
+                   [+a, +b, -c, 0],
+                   [+a, +c, 0, +b],
+                   [+a, +c, 0, -b],
+                   [+a, -b, +c, 0],
+                   [+a, -b, -c, 0],
+                   [+a, -c, 0, +b],
+                   [+a, -c, 0, -b],
+                   [+a, 0, +b, +c],
+                   [+a, 0, +b, -c],
+                   [+a, 0, -b, +c],
+                   [+a, 0, -b, -c],
+                   [+b, +a, 0, +c],
+                   [+b, +a, 0, -c],
+                   [+b, +c, +a, 0],
+                   [+b, +c, -a, 0],
+                   [+b, -a, 0, +c],
+                   [+b, -a, 0, -c],
+                   [+b, -c, +a, 0],
+                   [+b, -c, -a, 0],
+                   [+b, 0, +c, +a],
+                   [+b, 0, +c, -a],
+                   [+b, 0, -c, +a],
+                   [+b, 0, -c, -a],
+                   [+c, +a, +b, 0],
+                   [+c, +a, -b, 0],
+                   [+c, +b, 0, +a],
+                   [+c, +b, 0, -a],
+                   [+c, -a, +b, 0],
+                   [+c, -a, -b, 0],
+                   [+c, -b, 0, +a],
+                   [+c, -b, 0, -a],
+                   [+c, 0, +a, +b],
+                   [+c, 0, +a, -b],
+                   [+c, 0, -a, +b],
+                   [+c, 0, -a, -b],
+                   [0, +a, +c, +b],
+                   [0, +a, +c, -b],
+                   [0, +a, -c, +b],
+                   [0, +a, -c, -b],
+                   [0, +b, +a, +c],
+                   [0, +b, +a, -c],
+                   [0, +b, -a, +c],
+                   [0, +b, -a, -c],
+                   [0, +c, +b, +a],
+                   [0, +c, +b, -a],
+                   [0, +c, -b, +a],
+                   [0, +c, -b, -a]])
+    return cls.from_quat(np.concatenate((g1, g2)))
+
+
+def octahedral(cls):
+    g1 = tetrahedral(cls).as_quat()
+    c = np.sqrt(2) / 2
+    g2 = np.array([[+c, 0, 0, +c],
+                   [0, +c, 0, +c],
+                   [0, 0, +c, +c],
+                   [0, 0, -c, +c],
+                   [0, -c, 0, +c],
+                   [-c, 0, 0, +c],
+                   [0, +c, +c, 0],
+                   [0, -c, +c, 0],
+                   [+c, 0, +c, 0],
+                   [-c, 0, +c, 0],
+                   [+c, +c, 0, 0],
+                   [-c, +c, 0, 0]])
+    return cls.from_quat(np.concatenate((g1, g2)))
+
+
+def tetrahedral(cls):
+    g1 = np.eye(4)
+    c = 0.5
+    g2 = np.array([[c, -c, -c, +c],
+                   [c, -c, +c, +c],
+                   [c, +c, -c, +c],
+                   [c, +c, +c, +c],
+                   [c, -c, -c, -c],
+                   [c, -c, +c, -c],
+                   [c, +c, -c, -c],
+                   [c, +c, +c, -c]])
+    return cls.from_quat(np.concatenate((g1, g2)))
+
+
+def dicyclic(cls, n, axis=2):
+    g1 = cyclic(cls, n, axis).as_rotvec()
+
+    thetas = np.linspace(0, np.pi, n, endpoint=False)
+    rv = np.pi * np.vstack([np.zeros(n), np.cos(thetas), np.sin(thetas)]).T
+    g2 = np.roll(rv, axis, axis=1)
+    return cls.from_rotvec(np.concatenate((g1, g2)))
+
+
+def cyclic(cls, n, axis=2):
+    thetas = np.linspace(0, 2 * np.pi, n, endpoint=False)
+    rv = np.vstack([thetas, np.zeros(n), np.zeros(n)]).T
+    return cls.from_rotvec(np.roll(rv, axis, axis=1))
+
+
+def create_group(cls, group, axis='Z'):
+    if not isinstance(group, str):
+        raise ValueError("`group` argument must be a string")
+
+    permitted_axes = ['x', 'y', 'z', 'X', 'Y', 'Z']
+    if axis not in permitted_axes:
+        raise ValueError("`axis` must be one of " + ", ".join(permitted_axes))
+
+    if group in ['I', 'O', 'T']:
+        symbol = group
+        order = 1
+    elif group[:1] in ['C', 'D'] and group[1:].isdigit():
+        symbol = group[:1]
+        order = int(group[1:])
+    else:
+        raise ValueError("`group` must be one of 'I', 'O', 'T', 'Dn', 'Cn'")
+
+    if order < 1:
+        raise ValueError("Group order must be positive")
+
+    axis = 'xyz'.index(axis.lower())
+    if symbol == 'I':
+        return icosahedral(cls)
+    elif symbol == 'O':
+        return octahedral(cls)
+    elif symbol == 'T':
+        return tetrahedral(cls)
+    elif symbol == 'D':
+        return dicyclic(cls, order, axis=axis)
+    elif symbol == 'C':
+        return cyclic(cls, order, axis=axis)
+    else:
+        assert False

venv/Lib/site-packages/scipy/spatial/transform/_rotation_spline.py

@@ -0,0 +1,456 @@
+import numpy as np
+from scipy.linalg import solve_banded
+from .rotation import Rotation
+
+
+def _create_skew_matrix(x):
+    """Create skew-symmetric matrices corresponding to vectors.
+
+    Parameters
+    ----------
+    x : ndarray, shape (n, 3)
+        Set of vectors.
+
+    Returns
+    -------
+    ndarray, shape (n, 3, 3)
+    """
+    result = np.zeros((len(x), 3, 3))
+    result[:, 0, 1] = -x[:, 2]
+    result[:, 0, 2] = x[:, 1]
+    result[:, 1, 0] = x[:, 2]
+    result[:, 1, 2] = -x[:, 0]
+    result[:, 2, 0] = -x[:, 1]
+    result[:, 2, 1] = x[:, 0]
+    return result
+
+
+def _matrix_vector_product_of_stacks(A, b):
+    """Compute the product of stack of matrices and vectors."""
+    return np.einsum("ijk,ik->ij", A, b)
+
+
+def _angular_rate_to_rotvec_dot_matrix(rotvecs):
+    """Compute matrices to transform angular rates to rot. vector derivatives.
+
+    The matrices depend on the current attitude represented as a rotation
+    vector.
+
+    Parameters
+    ----------
+    rotvecs : ndarray, shape (n, 3)
+        Set of rotation vectors.
+
+    Returns
+    -------
+    ndarray, shape (n, 3, 3)
+    """
+    norm = np.linalg.norm(rotvecs, axis=1)
+    k = np.empty_like(norm)
+
+    mask = norm > 1e-4
+    nm = norm[mask]
+    k[mask] = (1 - 0.5 * nm / np.tan(0.5 * nm)) / nm**2
+    mask = ~mask
+    nm = norm[mask]
+    k[mask] = 1/12 + 1/720 * nm**2
+
+    skew = _create_skew_matrix(rotvecs)
+
+    result = np.empty((len(rotvecs), 3, 3))
+    result[:] = np.identity(3)
+    result[:] += 0.5 * skew
+    result[:] += k[:, None, None] * np.matmul(skew, skew)
+
+    return result
+
+
+def _rotvec_dot_to_angular_rate_matrix(rotvecs):
+    """Compute matrices to transform rot. vector derivatives to angular rates.
+
+    The matrices depend on the current attitude represented as a rotation
+    vector.
+
+    Parameters
+    ----------
+    rotvecs : ndarray, shape (n, 3)
+        Set of rotation vectors.
+
+    Returns
+    -------
+    ndarray, shape (n, 3, 3)
+    """
+    norm = np.linalg.norm(rotvecs, axis=1)
+    k1 = np.empty_like(norm)
+    k2 = np.empty_like(norm)
+
+    mask = norm > 1e-4
+    nm = norm[mask]
+    k1[mask] = (1 - np.cos(nm)) / nm ** 2
+    k2[mask] = (nm - np.sin(nm)) / nm ** 3
+
+    mask = ~mask
+    nm = norm[mask]
+    k1[mask] = 0.5 - nm ** 2 / 24
+    k2[mask] = 1 / 6 - nm ** 2 / 120
+
+    skew = _create_skew_matrix(rotvecs)
+
+    result = np.empty((len(rotvecs), 3, 3))
+    result[:] = np.identity(3)
+    result[:] -= k1[:, None, None] * skew
+    result[:] += k2[:, None, None] * np.matmul(skew, skew)
+
+    return result
+
+
+def _angular_acceleration_nonlinear_term(rotvecs, rotvecs_dot):
+    """Compute the non-linear term in angular acceleration.
+
+    The angular acceleration contains a quadratic term with respect to
+    the derivative of the rotation vector. This function computes that.
+
+    Parameters
+    ----------
+    rotvecs : ndarray, shape (n, 3)
+        Set of rotation vectors.
+    rotvecs_dot: ndarray, shape (n, 3)
+        Set of rotation vector derivatives.
+
+    Returns
+    -------
+    ndarray, shape (n, 3)
+    """
+    norm = np.linalg.norm(rotvecs, axis=1)
+    dp = np.sum(rotvecs * rotvecs_dot, axis=1)
+    cp = np.cross(rotvecs, rotvecs_dot)
+    ccp = np.cross(rotvecs, cp)
+    dccp = np.cross(rotvecs_dot, cp)
+
+    k1 = np.empty_like(norm)
+    k2 = np.empty_like(norm)
+    k3 = np.empty_like(norm)
+
+    mask = norm > 1e-4
+    nm = norm[mask]
+    k1[mask] = (-nm * np.sin(nm) - 2 * (np.cos(nm) - 1)) / nm ** 4
+    k2[mask] = (-2 * nm + 3 * np.sin(nm) - nm * np.cos(nm)) / nm ** 5
+    k3[mask] = (nm - np.sin(nm)) / nm ** 3
+
+    mask = ~mask
+    nm = norm[mask]
+    k1[mask] = 1/12 - nm ** 2 / 180
+    k2[mask] = -1/60 + nm ** 2 / 12604
+    k3[mask] = 1/6 - nm ** 2 / 120
+
+    dp = dp[:, None]
+    k1 = k1[:, None]
+    k2 = k2[:, None]
+    k3 = k3[:, None]
+
+    return dp * (k1 * cp + k2 * ccp) + k3 * dccp
+
+
+def _compute_angular_rate(rotvecs, rotvecs_dot):
+    """Compute angular rates given rotation vectors and its derivatives.
+
+    Parameters
+    ----------
+    rotvecs : ndarray, shape (n, 3)
+        Set of rotation vectors.
+    rotvecs_dot : ndarray, shape (n, 3)
+        Set of rotation vector derivatives.
+
+    Returns
+    -------
+    ndarray, shape (n, 3)
+    """
+    return _matrix_vector_product_of_stacks(
+        _rotvec_dot_to_angular_rate_matrix(rotvecs), rotvecs_dot)
+
+
+def _compute_angular_acceleration(rotvecs, rotvecs_dot, rotvecs_dot_dot):
+    """Compute angular acceleration given rotation vector and its derivatives.
+
+    Parameters
+    ----------
+    rotvecs : ndarray, shape (n, 3)
+        Set of rotation vectors.
+    rotvecs_dot : ndarray, shape (n, 3)
+        Set of rotation vector derivatives.
+    rotvecs_dot_dot : ndarray, shape (n, 3)
+        Set of rotation vector second derivatives.
+
+    Returns
+    -------
+    ndarray, shape (n, 3)
+    """
+    return (_compute_angular_rate(rotvecs, rotvecs_dot_dot) +
+            _angular_acceleration_nonlinear_term(rotvecs, rotvecs_dot))
+
+
+def _create_block_3_diagonal_matrix(A, B, d):
+    """Create a 3-diagonal block matrix as banded.
+
+    The matrix has the following structure:
+
+        DB...
+        ADB..
+        .ADB.
+        ..ADB
+        ...AD
+
+    The blocks A, B and D are 3-by-3 matrices. The D matrices has the form
+    d * I.
+
+    Parameters
+    ----------
+    A : ndarray, shape (n, 3, 3)
+        Stack of A blocks.
+    B : ndarray, shape (n, 3, 3)
+        Stack of B blocks.
+    d : ndarray, shape (n + 1,)
+        Values for diagonal blocks.
+
+    Returns
+    -------
+    ndarray, shape (11, 3 * (n + 1))
+        Matrix in the banded form as used by `scipy.linalg.solve_banded`.
+    """
+    ind = np.arange(3)
+    ind_blocks = np.arange(len(A))
+
+    A_i = np.empty_like(A, dtype=int)
+    A_i[:] = ind[:, None]
+    A_i += 3 * (1 + ind_blocks[:, None, None])
+
+    A_j = np.empty_like(A, dtype=int)
+    A_j[:] = ind
+    A_j += 3 * ind_blocks[:, None, None]
+
+    B_i = np.empty_like(B, dtype=int)
+    B_i[:] = ind[:, None]
+    B_i += 3 * ind_blocks[:, None, None]
+
+    B_j = np.empty_like(B, dtype=int)
+    B_j[:] = ind
+    B_j += 3 * (1 + ind_blocks[:, None, None])
+
+    diag_i = diag_j = np.arange(3 * len(d))
+    i = np.hstack((A_i.ravel(), B_i.ravel(), diag_i))
+    j = np.hstack((A_j.ravel(), B_j.ravel(), diag_j))
+    values = np.hstack((A.ravel(), B.ravel(), np.repeat(d, 3)))
+
+    u = 5
+    l = 5
+    result = np.zeros((u + l + 1, 3 * len(d)))
+    result[u + i - j, j] = values
+    return result
+
+
+class RotationSpline(object):
+    """Interpolate rotations with continuous angular rate and acceleration.
+
+    The rotation vectors between each consecutive orientation are cubic
+    functions of time and it is guaranteed that angular rate and acceleration
+    are continuous. Such interpolation are analogous to cubic spline
+    interpolation.
+
+    Refer to [1]_ for math and implementation details.
+
+    Parameters
+    ----------
+    times : array_like, shape (N,)
+        Times of the known rotations. At least 2 times must be specified.
+    rotations : `Rotation` instance
+        Rotations to perform the interpolation between. Must contain N
+        rotations.
+
+    Methods
+    -------
+    __call__
+
+    References
+    ----------
+    .. [1] `Smooth Attitude Interpolation
+            <https://github.com/scipy/scipy/files/2932755/attitude_interpolation.pdf>`_
+
+    Examples
+    --------
+    >>> from scipy.spatial.transform import Rotation, RotationSpline
+
+    Define the sequence of times and rotations from the Euler angles:
+
+    >>> times = [0, 10, 20, 40]
+    >>> angles = [[-10, 20, 30], [0, 15, 40], [-30, 45, 30], [20, 45, 90]]
+    >>> rotations = Rotation.from_euler('XYZ', angles, degrees=True)
+
+    Create the interpolator object:
+
+    >>> spline = RotationSpline(times, rotations)
+
+    Interpolate the Euler angles, angular rate and acceleration:
+
+    >>> angular_rate = np.rad2deg(spline(times, 1))
+    >>> angular_acceleration = np.rad2deg(spline(times, 2))
+    >>> times_plot = np.linspace(times[0], times[-1], 100)
+    >>> angles_plot = spline(times_plot).as_euler('XYZ', degrees=True)
+    >>> angular_rate_plot = np.rad2deg(spline(times_plot, 1))
+    >>> angular_acceleration_plot = np.rad2deg(spline(times_plot, 2))
+
+    On this plot you see that Euler angles are continuous and smooth:
+
+    >>> import matplotlib.pyplot as plt
+    >>> plt.plot(times_plot, angles_plot)
+    >>> plt.plot(times, angles, 'x')
+    >>> plt.title("Euler angles")
+    >>> plt.show()
+
+    The angular rate is also smooth:
+
+    >>> plt.plot(times_plot, angular_rate_plot)
+    >>> plt.plot(times, angular_rate, 'x')
+    >>> plt.title("Angular rate")
+    >>> plt.show()
+
+    The angular acceleration is continuous, but not smooth. Also note that
+    the angular acceleration is not a piecewise-linear function, because
+    it is different from the second derivative of the rotation vector (which
+    is a piecewise-linear function as in the cubic spline).
+
+    >>> plt.plot(times_plot, angular_acceleration_plot)
+    >>> plt.plot(times, angular_acceleration, 'x')
+    >>> plt.title("Angular acceleration")
+    >>> plt.show()
+    """
+    # Parameters for the solver for angular rate.
+    MAX_ITER = 10
+    TOL = 1e-9
+
+    def _solve_for_angular_rates(self, dt, angular_rates, rotvecs):
+        angular_rate_first = angular_rates[0].copy()
+
+        A = _angular_rate_to_rotvec_dot_matrix(rotvecs)
+        A_inv = _rotvec_dot_to_angular_rate_matrix(rotvecs)
+        M = _create_block_3_diagonal_matrix(
+            2 * A_inv[1:-1] / dt[1:-1, None, None],
+            2 * A[1:-1] / dt[1:-1, None, None],
+            4 * (1 / dt[:-1] + 1 / dt[1:]))
+
+        b0 = 6 * (rotvecs[:-1] * dt[:-1, None] ** -2 +
+                  rotvecs[1:] * dt[1:, None] ** -2)
+        b0[0] -= 2 / dt[0] * A_inv[0].dot(angular_rate_first)
+        b0[-1] -= 2 / dt[-1] * A[-1].dot(angular_rates[-1])
+
+        for iteration in range(self.MAX_ITER):
+            rotvecs_dot = _matrix_vector_product_of_stacks(A, angular_rates)
+            delta_beta = _angular_acceleration_nonlinear_term(
+                rotvecs[:-1], rotvecs_dot[:-1])
+            b = b0 - delta_beta
+            angular_rates_new = solve_banded((5, 5), M, b.ravel())
+            angular_rates_new = angular_rates_new.reshape((-1, 3))
+
+            delta = np.abs(angular_rates_new - angular_rates[:-1])
+            angular_rates[:-1] = angular_rates_new
+            if np.all(delta < self.TOL * (1 + np.abs(angular_rates_new))):
+                break
+
+        rotvecs_dot = _matrix_vector_product_of_stacks(A, angular_rates)
+        angular_rates = np.vstack((angular_rate_first, angular_rates[:-1]))
+
+        return angular_rates, rotvecs_dot
+
+    def __init__(self, times, rotations):
+        from scipy.interpolate import PPoly
+
+        if len(rotations) == 1:
+            raise ValueError("`rotations` must contain at least 2 rotations.")
+
+        times = np.asarray(times, dtype=float)
+        if times.ndim != 1:
+            raise ValueError("`times` must be 1-dimensional.")
+
+        if len(times) != len(rotations):
+            raise ValueError("Expected number of rotations to be equal to "
+                             "number of timestamps given, got {} rotations "
+                             "and {} timestamps."
+                             .format(len(rotations), len(times)))
+
+        dt = np.diff(times)
+        if np.any(dt <= 0):
+            raise ValueError("Values in `times` must be in a strictly "
+                             "increasing order.")
+
+        rotvecs = (rotations[:-1].inv() * rotations[1:]).as_rotvec()
+        angular_rates = rotvecs / dt[:, None]
+
+        if len(rotations) == 2:
+            rotvecs_dot = angular_rates
+        else:
+            angular_rates, rotvecs_dot = self._solve_for_angular_rates(
+                dt, angular_rates, rotvecs)
+
+        dt = dt[:, None]
+        coeff = np.empty((4, len(times) - 1, 3))
+        coeff[0] = (-2 * rotvecs + dt * angular_rates
+                    + dt * rotvecs_dot) / dt ** 3
+        coeff[1] = (3 * rotvecs - 2 * dt * angular_rates
+                    - dt * rotvecs_dot) / dt ** 2
+        coeff[2] = angular_rates
+        coeff[3] = 0
+
+        self.times = times
+        self.rotations = rotations
+        self.interpolator = PPoly(coeff, times)
+
+    def __call__(self, times, order=0):
+        """Compute interpolated values.
+
+        Parameters
+        ----------
+        times : float or array_like
+            Times of interest.
+        order : {0, 1, 2}, optional
+            Order of differentiation:
+
+                * 0 (default) : return Rotation
+                * 1 : return the angular rate in rad/sec
+                * 2 : return the angular acceleration in rad/sec/sec
+
+        Returns
+        -------
+        Interpolated Rotation, angular rate or acceleration.
+        """
+        if order not in [0, 1, 2]:
+            raise ValueError("`order` must be 0, 1 or 2.")
+
+        times = np.asarray(times, dtype=float)
+        if times.ndim > 1:
+            raise ValueError("`times` must be at most 1-dimensional.")
+
+        singe_time = times.ndim == 0
+        times = np.atleast_1d(times)
+
+        rotvecs = self.interpolator(times)
+        if order == 0:
+            index = np.searchsorted(self.times, times, side='right')
+            index -= 1
+            index[index < 0] = 0
+            n_segments = len(self.times) - 1
+            index[index > n_segments - 1] = n_segments - 1
+            result = self.rotations[index] * Rotation.from_rotvec(rotvecs)
+        elif order == 1:
+            rotvecs_dot = self.interpolator(times, 1)
+            result = _compute_angular_rate(rotvecs, rotvecs_dot)
+        elif order == 2:
+            rotvecs_dot = self.interpolator(times, 1)
+            rotvecs_dot_dot = self.interpolator(times, 2)
+            result = _compute_angular_acceleration(rotvecs, rotvecs_dot,
+                                                   rotvecs_dot_dot)
+        else:
+            assert False
+
+        if singe_time:
+            result = result[0]
+
+        return result

venv/Lib/site-packages/scipy/spatial/transform/setup.py

@@ -0,0 +1,9 @@
+
+def configuration(parent_package='', top_path=None):
+    from numpy.distutils.misc_util import Configuration
+
+    config = Configuration('transform', parent_package, top_path)
+
+    config.add_data_dir('tests')
+
+    return config

venv/Lib/site-packages/scipy/spatial/transform/tests/test_rotation_groups.py

@@ -0,0 +1,169 @@
+import pytest
+
+import numpy as np
+from numpy.testing import assert_array_almost_equal
+from scipy.spatial.transform import Rotation
+from scipy.optimize import linear_sum_assignment
+from scipy.spatial.distance import cdist
+from scipy.constants import golden as phi
+from scipy.spatial import cKDTree
+
+
+TOL = 1E-12
+NS = range(1, 13)
+NAMES = ["I", "O", "T"] + ["C%d" % n for n in NS] + ["D%d" % n for n in NS]
+SIZES = [60, 24, 12] + list(NS) + [2 * n for n in NS]
+
+
+def _calculate_rmsd(P, Q):
+    """Calculates the root-mean-square distance between the points of P and Q.
+    The distance is taken as the minimum over all possible matchings. It is
+    zero if P and Q are identical and non-zero if not.
+    """
+    distance_matrix = cdist(P, Q, metric='sqeuclidean')
+    matching = linear_sum_assignment(distance_matrix)
+    return np.sqrt(distance_matrix[matching].sum())
+
+
+def _generate_pyramid(n, axis):
+    thetas = np.linspace(0, 2 * np.pi, n + 1)[:-1]
+    P = np.vstack([np.zeros(n), np.cos(thetas), np.sin(thetas)]).T
+    P = np.concatenate((P, [[1, 0, 0]]))
+    return np.roll(P, axis, axis=1)
+
+
+def _generate_prism(n, axis):
+    thetas = np.linspace(0, 2 * np.pi, n + 1)[:-1]
+    bottom = np.vstack([-np.ones(n), np.cos(thetas), np.sin(thetas)]).T
+    top = np.vstack([+np.ones(n), np.cos(thetas), np.sin(thetas)]).T
+    P = np.concatenate((bottom, top))
+    return np.roll(P, axis, axis=1)
+
+
+def _generate_icosahedron():
+    x = np.array([[0, -1, -phi],
+                  [0, -1, +phi],
+                  [0, +1, -phi],
+                  [0, +1, +phi]])
+    return np.concatenate([np.roll(x, i, axis=1) for i in range(3)])
+
+
+def _generate_octahedron():
+    return np.array([[-1, 0, 0], [+1, 0, 0], [0, -1, 0],
+                     [0, +1, 0], [0, 0, -1], [0, 0, +1]])
+
+
+def _generate_tetrahedron():
+    return np.array([[1, 1, 1], [1, -1, -1], [-1, 1, -1], [-1, -1, 1]])
+
+
+@pytest.mark.parametrize("name", [-1, None, True, np.array(['C3'])])
+def test_group_type(name):
+    with pytest.raises(ValueError,
+                       match="must be a string"):
+        Rotation.create_group(name)
+
+
+@pytest.mark.parametrize("name", ["Q", " ", "CA", "C ", "DA", "D ", "I2", ""])
+def test_group_name(name):
+    with pytest.raises(ValueError,
+                       match="must be one of 'I', 'O', 'T', 'Dn', 'Cn'"):
+        Rotation.create_group(name)
+
+
+@pytest.mark.parametrize("name", ["C0", "D0"])
+def test_group_order_positive(name):
+    with pytest.raises(ValueError,
+                       match="Group order must be positive"):
+        Rotation.create_group(name)
+
+
+@pytest.mark.parametrize("axis", ['A', 'b', 0, 1, 2, 4, False, None])
+def test_axis_valid(axis):
+    with pytest.raises(ValueError,
+                       match="`axis` must be one of"):
+        Rotation.create_group("C1", axis)
+
+
+def test_icosahedral():
+    """The icosahedral group fixes the rotations of an icosahedron. Here we
+    test that the icosahedron is invariant after application of the elements
+    of the rotation group."""
+    P = _generate_icosahedron()
+    for g in Rotation.create_group("I"):
+        g = Rotation.from_quat(g.as_quat())
+        assert _calculate_rmsd(P, g.apply(P)) < TOL
+
+
+def test_octahedral():
+    """Test that the octahedral group correctly fixes the rotations of an
+    octahedron."""
+    P = _generate_octahedron()
+    for g in Rotation.create_group("O"):
+        assert _calculate_rmsd(P, g.apply(P)) < TOL
+
+
+def test_tetrahedral():
+    """Test that the tetrahedral group correctly fixes the rotations of a
+    tetrahedron."""
+    P = _generate_tetrahedron()
+    for g in Rotation.create_group("T"):
+        assert _calculate_rmsd(P, g.apply(P)) < TOL
+
+
+@pytest.mark.parametrize("n", NS)
+@pytest.mark.parametrize("axis", 'XYZ')
+def test_dicyclic(n, axis):
+    """Test that the dicyclic group correctly fixes the rotations of a
+    prism."""
+    P = _generate_prism(n, axis='XYZ'.index(axis))
+    for g in Rotation.create_group("D%d" % n, axis=axis):
+        assert _calculate_rmsd(P, g.apply(P)) < TOL
+
+
+@pytest.mark.parametrize("n", NS)
+@pytest.mark.parametrize("axis", 'XYZ')
+def test_cyclic(n, axis):
+    """Test that the cyclic group correctly fixes the rotations of a
+    pyramid."""
+    P = _generate_pyramid(n, axis='XYZ'.index(axis))
+    for g in Rotation.create_group("C%d" % n, axis=axis):
+        assert _calculate_rmsd(P, g.apply(P)) < TOL
+
+
+@pytest.mark.parametrize("name, size", zip(NAMES, SIZES))
+def test_group_sizes(name, size):
+    assert len(Rotation.create_group(name)) == size
+
+
+@pytest.mark.parametrize("name, size", zip(NAMES, SIZES))
+def test_group_no_duplicates(name, size):
+    g = Rotation.create_group(name)
+    kdtree = cKDTree(g.as_quat())
+    assert len(kdtree.query_pairs(1E-3)) == 0
+
+
+@pytest.mark.parametrize("name, size", zip(NAMES, SIZES))
+def test_group_symmetry(name, size):
+    g = Rotation.create_group(name)
+    q = np.concatenate((-g.as_quat(), g.as_quat()))
+    distance = np.sort(cdist(q, q))
+    deltas = np.max(distance, axis=0) - np.min(distance, axis=0)
+    assert (deltas < TOL).all()
+
+
+@pytest.mark.parametrize("name", NAMES)
+def test_reduction(name):
+    """Test that the elements of the rotation group are correctly
+    mapped onto the identity rotation."""
+    g = Rotation.create_group(name)
+    f = g.reduce(g)
+    assert_array_almost_equal(f.magnitude(), np.zeros(len(g)))
+
+
+@pytest.mark.parametrize("name", NAMES)
+def test_single_reduction(name):
+    g = Rotation.create_group(name)
+    f = g[-1].reduce(g)
+    assert_array_almost_equal(f.magnitude(), 0)
+    assert f.as_quat().shape == (4,)

venv/Lib/site-packages/scipy/spatial/transform/tests/test_rotation_spline.py

@@ -0,0 +1,161 @@
+from itertools import product
+import numpy as np
+from numpy.testing import assert_allclose
+from pytest import raises
+from scipy.spatial.transform import Rotation, RotationSpline
+from scipy.spatial.transform._rotation_spline import (
+    _angular_rate_to_rotvec_dot_matrix,
+    _rotvec_dot_to_angular_rate_matrix,
+    _matrix_vector_product_of_stacks,
+    _angular_acceleration_nonlinear_term,
+    _create_block_3_diagonal_matrix)
+
+
+def test_angular_rate_to_rotvec_conversions():
+    np.random.seed(0)
+    rv = np.random.randn(4, 3)
+    A = _angular_rate_to_rotvec_dot_matrix(rv)
+    A_inv = _rotvec_dot_to_angular_rate_matrix(rv)
+
+    # When the rotation vector is aligned with the angular rate, then
+    # the rotation vector rate and angular rate are the same.
+    assert_allclose(_matrix_vector_product_of_stacks(A, rv), rv)
+    assert_allclose(_matrix_vector_product_of_stacks(A_inv, rv), rv)
+
+    # A and A_inv must be reciprocal to each other.
+    I_stack = np.empty((4, 3, 3))
+    I_stack[:] = np.eye(3)
+    assert_allclose(np.matmul(A, A_inv), I_stack, atol=1e-15)
+
+
+def test_angular_rate_nonlinear_term():
+    # The only simple test is to check that the term is zero when
+    # the rotation vector
+    np.random.seed(0)
+    rv = np.random.rand(4, 3)
+    assert_allclose(_angular_acceleration_nonlinear_term(rv, rv), 0,
+                    atol=1e-19)
+
+
+def test_create_block_3_diagonal_matrix():
+    np.random.seed(0)
+    A = np.empty((4, 3, 3))
+    A[:] = np.arange(1, 5)[:, None, None]
+
+    B = np.empty((4, 3, 3))
+    B[:] = -np.arange(1, 5)[:, None, None]
+    d = 10 * np.arange(10, 15)
+
+    banded = _create_block_3_diagonal_matrix(A, B, d)
+
+    # Convert the banded matrix to the full matrix.
+    k, l = list(zip(*product(np.arange(banded.shape[0]),
+                             np.arange(banded.shape[1]))))
+    k = np.asarray(k)
+    l = np.asarray(l)
+
+    i = k - 5 + l
+    j = l
+    values = banded.ravel()
+    mask = (i >= 0) & (i < 15)
+    i = i[mask]
+    j = j[mask]
+    values = values[mask]
+    full = np.zeros((15, 15))
+    full[i, j] = values
+
+    zero = np.zeros((3, 3))
+    eye = np.eye(3)
+
+    # Create the reference full matrix in the most straightforward manner.
+    ref = np.block([
+        [d[0] * eye, B[0], zero, zero, zero],
+        [A[0], d[1] * eye, B[1], zero, zero],
+        [zero, A[1], d[2] * eye, B[2], zero],
+        [zero, zero, A[2], d[3] * eye, B[3]],
+        [zero, zero, zero, A[3], d[4] * eye],
+    ])
+
+    assert_allclose(full, ref, atol=1e-19)
+
+
+def test_spline_2_rotations():
+    times = [0, 10]
+    rotations = Rotation.from_euler('xyz', [[0, 0, 0], [10, -20, 30]],
+                                    degrees=True)
+    spline = RotationSpline(times, rotations)
+
+    rv = (rotations[0].inv() * rotations[1]).as_rotvec()
+    rate = rv / (times[1] - times[0])
+    times_check = np.array([-1, 5, 12])
+    dt = times_check - times[0]
+    rv_ref = rate * dt[:, None]
+
+    assert_allclose(spline(times_check).as_rotvec(), rv_ref)
+    assert_allclose(spline(times_check, 1), np.resize(rate, (3, 3)))
+    assert_allclose(spline(times_check, 2), 0, atol=1e-16)
+
+
+def test_constant_attitude():
+    times = np.arange(10)
+    rotations = Rotation.from_rotvec(np.ones((10, 3)))
+    spline = RotationSpline(times, rotations)
+
+    times_check = np.linspace(-1, 11)
+    assert_allclose(spline(times_check).as_rotvec(), 1, rtol=1e-15)
+    assert_allclose(spline(times_check, 1), 0, atol=1e-19)
+    assert_allclose(spline(times_check, 2), 0, atol=1e-19)
+
+    assert_allclose(spline(5.5).as_rotvec(), 1, rtol=1e-15)
+    assert_allclose(spline(5.5, 1), 0, atol=1e-19)
+    assert_allclose(spline(5.5, 2), 0, atol=1e-19)
+
+
+def test_spline_properties():
+    times = np.array([0, 5, 15, 27])
+    angles = [[-5, 10, 27], [3, 5, 38], [-12, 10, 25], [-15, 20, 11]]
+
+    rotations = Rotation.from_euler('xyz', angles, degrees=True)
+    spline = RotationSpline(times, rotations)
+
+    assert_allclose(spline(times).as_euler('xyz', degrees=True), angles)
+    assert_allclose(spline(0).as_euler('xyz', degrees=True), angles[0])
+
+    h = 1e-8
+    rv0 = spline(times).as_rotvec()
+    rvm = spline(times - h).as_rotvec()
+    rvp = spline(times + h).as_rotvec()
+    assert_allclose(rv0, 0.5 * (rvp + rvm), rtol=1e-15)
+
+    r0 = spline(times, 1)
+    rm = spline(times - h, 1)
+    rp = spline(times + h, 1)
+    assert_allclose(r0, 0.5 * (rm + rp), rtol=1e-14)
+
+    a0 = spline(times, 2)
+    am = spline(times - h, 2)
+    ap = spline(times + h, 2)
+    assert_allclose(a0, am, rtol=1e-7)
+    assert_allclose(a0, ap, rtol=1e-7)
+
+
+def test_error_handling():
+    raises(ValueError, RotationSpline, [1.0], Rotation.random())
+
+    r = Rotation.random(10)
+    t = np.arange(10).reshape(5, 2)
+    raises(ValueError, RotationSpline, t, r)
+
+    t = np.arange(9)
+    raises(ValueError, RotationSpline, t, r)
+
+    t = np.arange(10)
+    t[5] = 0
+    raises(ValueError, RotationSpline, t, r)
+
+    t = np.arange(10)
+
+    s = RotationSpline(t, r)
+    raises(ValueError, s, 10, -1)
+
+    raises(ValueError, s, np.arange(10).reshape(5, 2))