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