interpolate

Overview

`rowan.interpolate.slerp`	Spherical linear interpolation between p and q.
`rowan.interpolate.slerp_prime`	Compute the derivative of slerp.
`rowan.interpolate.squad`	Cubically interpolate between p and q.

Details

Interpolate between pairs of quaternions.

The rowan package provides a simple interface to slerp, the standard method of quaternion interpolation for two quaternions.

rowan.interpolate.slerp(q0, q1, t, ensure_shortest=True)

Spherical linear interpolation between p and q.

The slerp formula can be easily expressed in terms of the quaternion exponential (see rowan.exp()).

Parameters:

q0 ((…, 4) numpy.ndarray) – First array of quaternions.
q1 ((…, 4) numpy.ndarray) – Second array of quaternions.
t ((…) numpy.ndarray) – Interpolation parameter \(\in [0, 1]\)
ensure_shortest (bool) – Flip quaternions to ensure we traverse the geodesic in the shorter (\(<180^{\circ}\)) direction.

Note

Given inputs such that \(t\notin [0, 1]\), the values outside the range are simply assumed to be 0 or 1 (depending on which side of the interval they fall on).

Returns:: Interpolations between p and q.
Return type:: (…, 4) numpy.ndarray

Example:

>>> import numpy as np
>>> rowan.interpolate.slerp(
...     [[1, 0, 0, 0]], [[np.sqrt(2)/2, np.sqrt(2)/2, 0, 0]], 0.5)
array([[0.92387953, 0.38268343, 0.        , 0.        ]])

rowan.interpolate.slerp_prime(q0, q1, t, ensure_shortest=True)

Compute the derivative of slerp.

Parameters:

q0 ((…, 4) numpy.ndarray) – First set of quaternions.
q1 ((…, 4) numpy.ndarray) – Second set of quaternions.
t ((…) numpy.ndarray) – Interpolation parameter \(\in [0, 1]\)
ensure_shortest (bool) – Flip quaternions to ensure we traverse the geodesic in the shorter (\(<180^{\circ}\)) direction

Returns:

The derivative of the interpolations between p and q.

Return type:

(…, 4) numpy.ndarray

Example:

import numpy as np
q_slerp_prime rowan.interpolate.slerp_prime(
    [[1, 0, 0, 0]], [[np.sqrt(2)/2, np.sqrt(2)/2, 0, 0]], 0.5)

rowan.interpolate.squad(p, a, b, q, t)

Cubically interpolate between p and q.

The SQUAD formula is just a repeated application of Slerp between multiple quaternions as originally derived in [Shoemake85]:

\[\begin{equation} \textrm{squad}(p, a, b, q, t) = \textrm{slerp}(p, q, t) \left(\textrm{slerp}(p, q, t)^{-1}\textrm{slerp}(a, b, t) \right)^{2t(1-t)} \end{equation}\]

[Shoemake85]

Ken Shoemake. Animating rotation with quaternion curves. SIGGRAPH Comput. Graph., 19(3):245-254, July 1985.

Parameters:

p ((…, 4) numpy.ndarray) – First endpoint of interpolation.
a ((…, 4) numpy.ndarray) – First control point of interpolation.
b ((…, 4) numpy.ndarray) – Second control point of interpolation.
q ((…, 4) numpy.ndarray) – Second endpoint of interpolation.
t ((…) numpy.ndarray) – Interpolation parameter \(t \in [0, 1]\).

Returns:

Interpolations between p and q.

Return type:

(…, 4) numpy.ndarray

Example:

>>> import numpy as np
>>> rowan.interpolate.squad(
...     [1, 0, 0, 0], [np.sqrt(2)/2, np.sqrt(2)/2, 0, 0],
...     [0, np.sqrt(2)/2, np.sqrt(2)/2, 0],
...     [0, 0, np.sqrt(2)/2, np.sqrt(2)/2], 0.5)
array([[0.64550177, 0.47254009, 0.52564058, 0.28937053]])