papyri
{ } Raw JSON

bundles / scipy latest / scipy / spatial / _geometric_slerp / geometric_slerp

function

scipy.spatial._geometric_slerp:geometric_slerp

source: /scipy/spatial/_geometric_slerp.py :33

Signature

def   geometric_slerp ( start : npt.ArrayLike ,  end : npt.ArrayLike ,  t : npt.ArrayLike ,  tol : float = 1e-07 )  →  np.ndarray

Summary

Geometric spherical linear interpolation.

Extended Summary

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. 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 or end are 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].

Examples

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()
✓
fig-bb8878fe8231df10.png
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: 
import warnings
opposite_pole = np.array([-1, 0])
✓
with warnings.catch_warnings():
    warnings.simplefilter("ignore", UserWarning)
    geometric_slerp(start,
                    opposite_pole,
                    t_vals)
✗
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()
✓
fig-629bd5fc1e3697da.png
It is also possible to perform extrapolations outside the interpolation interval by using interpolation parameter values below 0 or above 1. For example, the above example may be adjusted to extrapolate to an antipodal position: 
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
✓
ax.plot_surface(x, y, z, color='y', alpha=0.1)
✗
start = np.array([1, 0, 0])
end = np.array([0, 0, 1])
t_vals = np.linspace(0, 2, 400)
result = geometric_slerp(start,
                         end,
                         t_vals)
✓
ax.plot(result[...,0], result[...,1], result[...,2], c='k')
✗
plt.show()
✓
fig-588288dfae26969e.png

See also

scipy.spatial.transform.Slerp

3-D Slerp that works with quaternions

Aliases

  • scipy.spatial.geometric_slerp

Referenced by

This package