venv/lib/python3.10/site-packages/scipy/spatial/_geometric_slerp.py

from __future__ import annotations

__all__ = ['geometric_slerp']

import warnings
from typing import TYPE_CHECKING

import numpy as np
from scipy.spatial.distance import euclidean

if TYPE_CHECKING:
    import numpy.typing as npt


def _geometric_slerp(start, end, t):
    # create an orthogonal basis using QR decomposition
    basis = np.vstack([start, end])
    Q, R = np.linalg.qr(basis.T)
    signs = 2 * (np.diag(R) >= 0) - 1
    Q = Q.T * signs.T[:, np.newaxis]
    R = R.T * signs.T[:, np.newaxis]

    # calculate the angle between `start` and `end`
    c = np.dot(start, end)
    s = np.linalg.det(R)
    omega = np.arctan2(s, c)

    # interpolate
    start, end = Q
    s = np.sin(t * omega)
    c = np.cos(t * omega)
    return start * c[:, np.newaxis] + end * s[:, np.newaxis]


def geometric_slerp(
    start: npt.ArrayLike,
    end: npt.ArrayLike,
    t: npt.ArrayLike,
    tol: float = 1e-7,
) -> np.ndarray:
    """
    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.

    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]_.

    .. 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.

    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()

    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()
    """

    start = np.asarray(start, dtype=np.float64)
    end = np.asarray(end, dtype=np.float64)
    t = np.asarray(t)

    if t.ndim > 1:
        raise ValueError("The interpolation parameter "
                         "value must be one dimensional.")

    if start.ndim != 1 or end.ndim != 1:
        raise ValueError("Start and end coordinates "
                         "must be one-dimensional")

    if start.size != end.size:
        raise ValueError("The dimensions of start and "
                         "end must match (have same size)")

    if start.size < 2 or end.size < 2:
        raise ValueError("The start and end coordinates must "
                         "both be in at least two-dimensional "
                         "space")

    if np.array_equal(start, end):
        return np.linspace(start, start, t.size)

    # for points that violate equation for n-sphere
    for coord in [start, end]:
        if not np.allclose(np.linalg.norm(coord), 1.0,
                           rtol=1e-9,
                           atol=0):
            raise ValueError("start and end are not"
                             " on a unit n-sphere")

    if not isinstance(tol, float):
        raise ValueError("tol must be a float")
    else:
        tol = np.fabs(tol)

    coord_dist = euclidean(start, end)

    # diameter of 2 within tolerance means antipodes, which is a problem
    # for all unit n-spheres (even the 0-sphere would have an ambiguous path)
    if np.allclose(coord_dist, 2.0, rtol=0, atol=tol):
        warnings.warn("start and end are antipodes "
                      "using the specified tolerance; "
                      "this may cause ambiguous slerp paths",
                      stacklevel=2)

    t = np.asarray(t, dtype=np.float64)

    if t.size == 0:
        return np.empty((0, start.size))

    if t.min() < 0 or t.max() > 1:
        raise ValueError("interpolation parameter must be in [0, 1]")

    if t.ndim == 0:
        return _geometric_slerp(start,
                                end,
                                np.atleast_1d(t)).ravel()
    else:
        return _geometric_slerp(start,
                                end,
                                t)