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