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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.
t: float 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.
tol: float
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.
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]_.
.. versionadded:: 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.
See Also
--------
scipy.spatial.transform.Slerp : 3-D Slerp that works with quaternions
Examples
--------
Interpolate four linearly-spaced values on the circumference of
a circle spanning 90 degrees:
>>> 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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