263 lines
10 KiB
Python
263 lines
10 KiB
Python
"""
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Tools for triangular grids.
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"""
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import numpy as np
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from matplotlib import cbook
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from matplotlib.tri import Triangulation
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class TriAnalyzer:
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"""
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Define basic tools for triangular mesh analysis and improvement.
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A TriAnalyzer encapsulates a `.Triangulation` object and provides basic
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tools for mesh analysis and mesh improvement.
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Attributes
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----------
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scale_factors
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Parameters
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----------
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triangulation : `~matplotlib.tri.Triangulation`
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The encapsulated triangulation to analyze.
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"""
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def __init__(self, triangulation):
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cbook._check_isinstance(Triangulation, triangulation=triangulation)
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self._triangulation = triangulation
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@property
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def scale_factors(self):
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"""
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Factors to rescale the triangulation into a unit square.
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Returns
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-------
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(float, float)
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Scaling factors (kx, ky) so that the triangulation
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``[triangulation.x * kx, triangulation.y * ky]``
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fits exactly inside a unit square.
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"""
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compressed_triangles = self._triangulation.get_masked_triangles()
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node_used = (np.bincount(np.ravel(compressed_triangles),
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minlength=self._triangulation.x.size) != 0)
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return (1 / np.ptp(self._triangulation.x[node_used]),
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1 / np.ptp(self._triangulation.y[node_used]))
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def circle_ratios(self, rescale=True):
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"""
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Return a measure of the triangulation triangles flatness.
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The ratio of the incircle radius over the circumcircle radius is a
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widely used indicator of a triangle flatness.
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It is always ``<= 0.5`` and ``== 0.5`` only for equilateral
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triangles. Circle ratios below 0.01 denote very flat triangles.
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To avoid unduly low values due to a difference of scale between the 2
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axis, the triangular mesh can first be rescaled to fit inside a unit
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square with `scale_factors` (Only if *rescale* is True, which is
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its default value).
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Parameters
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----------
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rescale : bool, default: True
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If True, internally rescale (based on `scale_factors`), so that the
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(unmasked) triangles fit exactly inside a unit square mesh.
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Returns
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-------
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masked array
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Ratio of the incircle radius over the circumcircle radius, for
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each 'rescaled' triangle of the encapsulated triangulation.
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Values corresponding to masked triangles are masked out.
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"""
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# Coords rescaling
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if rescale:
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(kx, ky) = self.scale_factors
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else:
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(kx, ky) = (1.0, 1.0)
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pts = np.vstack([self._triangulation.x*kx,
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self._triangulation.y*ky]).T
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tri_pts = pts[self._triangulation.triangles]
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# Computes the 3 side lengths
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a = tri_pts[:, 1, :] - tri_pts[:, 0, :]
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b = tri_pts[:, 2, :] - tri_pts[:, 1, :]
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c = tri_pts[:, 0, :] - tri_pts[:, 2, :]
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a = np.hypot(a[:, 0], a[:, 1])
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b = np.hypot(b[:, 0], b[:, 1])
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c = np.hypot(c[:, 0], c[:, 1])
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# circumcircle and incircle radii
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s = (a+b+c)*0.5
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prod = s*(a+b-s)*(a+c-s)*(b+c-s)
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# We have to deal with flat triangles with infinite circum_radius
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bool_flat = (prod == 0.)
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if np.any(bool_flat):
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# Pathologic flow
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ntri = tri_pts.shape[0]
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circum_radius = np.empty(ntri, dtype=np.float64)
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circum_radius[bool_flat] = np.inf
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abc = a*b*c
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circum_radius[~bool_flat] = abc[~bool_flat] / (
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4.0*np.sqrt(prod[~bool_flat]))
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else:
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# Normal optimized flow
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circum_radius = (a*b*c) / (4.0*np.sqrt(prod))
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in_radius = (a*b*c) / (4.0*circum_radius*s)
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circle_ratio = in_radius/circum_radius
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mask = self._triangulation.mask
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if mask is None:
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return circle_ratio
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else:
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return np.ma.array(circle_ratio, mask=mask)
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def get_flat_tri_mask(self, min_circle_ratio=0.01, rescale=True):
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"""
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Eliminate excessively flat border triangles from the triangulation.
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Returns a mask *new_mask* which allows to clean the encapsulated
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triangulation from its border-located flat triangles
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(according to their :meth:`circle_ratios`).
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This mask is meant to be subsequently applied to the triangulation
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using `.Triangulation.set_mask`.
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*new_mask* is an extension of the initial triangulation mask
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in the sense that an initially masked triangle will remain masked.
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The *new_mask* array is computed recursively; at each step flat
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triangles are removed only if they share a side with the current mesh
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border. Thus no new holes in the triangulated domain will be created.
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Parameters
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----------
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min_circle_ratio : float, default: 0.01
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Border triangles with incircle/circumcircle radii ratio r/R will
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be removed if r/R < *min_circle_ratio*.
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rescale : bool, default: True
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If True, first, internally rescale (based on `scale_factors`) so
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that the (unmasked) triangles fit exactly inside a unit square
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mesh. This rescaling accounts for the difference of scale which
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might exist between the 2 axis.
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Returns
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-------
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bool array-like
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Mask to apply to encapsulated triangulation.
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All the initially masked triangles remain masked in the
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*new_mask*.
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Notes
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-----
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The rationale behind this function is that a Delaunay
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triangulation - of an unstructured set of points - sometimes contains
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almost flat triangles at its border, leading to artifacts in plots
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(especially for high-resolution contouring).
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Masked with computed *new_mask*, the encapsulated
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triangulation would contain no more unmasked border triangles
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with a circle ratio below *min_circle_ratio*, thus improving the
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mesh quality for subsequent plots or interpolation.
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"""
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# Recursively computes the mask_current_borders, true if a triangle is
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# at the border of the mesh OR touching the border through a chain of
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# invalid aspect ratio masked_triangles.
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ntri = self._triangulation.triangles.shape[0]
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mask_bad_ratio = self.circle_ratios(rescale) < min_circle_ratio
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current_mask = self._triangulation.mask
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if current_mask is None:
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current_mask = np.zeros(ntri, dtype=bool)
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valid_neighbors = np.copy(self._triangulation.neighbors)
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renum_neighbors = np.arange(ntri, dtype=np.int32)
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nadd = -1
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while nadd != 0:
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# The active wavefront is the triangles from the border (unmasked
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# but with a least 1 neighbor equal to -1
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wavefront = (np.min(valid_neighbors, axis=1) == -1) & ~current_mask
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# The element from the active wavefront will be masked if their
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# circle ratio is bad.
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added_mask = wavefront & mask_bad_ratio
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current_mask = added_mask | current_mask
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nadd = np.sum(added_mask)
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# now we have to update the tables valid_neighbors
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valid_neighbors[added_mask, :] = -1
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renum_neighbors[added_mask] = -1
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valid_neighbors = np.where(valid_neighbors == -1, -1,
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renum_neighbors[valid_neighbors])
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return np.ma.filled(current_mask, True)
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def _get_compressed_triangulation(self):
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"""
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Compress (if masked) the encapsulated triangulation.
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Returns minimal-length triangles array (*compressed_triangles*) and
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coordinates arrays (*compressed_x*, *compressed_y*) that can still
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describe the unmasked triangles of the encapsulated triangulation.
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Returns
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-------
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compressed_triangles : array-like
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the returned compressed triangulation triangles
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compressed_x : array-like
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the returned compressed triangulation 1st coordinate
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compressed_y : array-like
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the returned compressed triangulation 2nd coordinate
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tri_renum : int array
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renumbering table to translate the triangle numbers from the
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encapsulated triangulation into the new (compressed) renumbering.
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-1 for masked triangles (deleted from *compressed_triangles*).
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node_renum : int array
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renumbering table to translate the point numbers from the
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encapsulated triangulation into the new (compressed) renumbering.
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-1 for unused points (i.e. those deleted from *compressed_x* and
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*compressed_y*).
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"""
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# Valid triangles and renumbering
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tri_mask = self._triangulation.mask
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compressed_triangles = self._triangulation.get_masked_triangles()
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ntri = self._triangulation.triangles.shape[0]
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if tri_mask is not None:
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tri_renum = self._total_to_compress_renum(~tri_mask)
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else:
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tri_renum = np.arange(ntri, dtype=np.int32)
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# Valid nodes and renumbering
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valid_node = (np.bincount(np.ravel(compressed_triangles),
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minlength=self._triangulation.x.size) != 0)
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compressed_x = self._triangulation.x[valid_node]
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compressed_y = self._triangulation.y[valid_node]
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node_renum = self._total_to_compress_renum(valid_node)
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# Now renumbering the valid triangles nodes
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compressed_triangles = node_renum[compressed_triangles]
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return (compressed_triangles, compressed_x, compressed_y, tri_renum,
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node_renum)
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@staticmethod
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def _total_to_compress_renum(valid):
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"""
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Parameters
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----------
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valid : 1d bool array
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Validity mask.
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Returns
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-------
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int array
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Array so that (`valid_array` being a compressed array
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based on a `masked_array` with mask ~*valid*):
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- For all i with valid[i] = True:
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valid_array[renum[i]] = masked_array[i]
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- For all i with valid[i] = False:
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renum[i] = -1 (invalid value)
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"""
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renum = np.full(np.size(valid), -1, dtype=np.int32)
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n_valid = np.sum(valid)
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renum[valid] = np.arange(n_valid, dtype=np.int32)
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return renum
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