scipy.spatial.geometric_slerp

scipy.spatial.geometric_slerp(start, end, t, tol=1e-07)

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.

tfloat or (n_points,) 1D array-like

A float or 1D 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.

tolfloat

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.

See also

scipy.spatial.transform.Slerp

3-D Slerp that works with quaternions

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].

Added in version 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.

Examples

Try it in your browser!

Interpolate four linearly-spaced values on the circumference of a circle spanning 90 degrees:

>>> import numpy as np
>>> 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()

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