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,) array-like A float or 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()