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