ftj! tdgZddlZddlmZddlZddlmZerddlmZ dZ dddd dd dd e d ej f d Z dS)geometric_slerpN) TYPE_CHECKING) euclideanctj||g}tj|j\}}dtj|dkzdz }|j|jddtjfz}|j|jddtjfz}tj||}tj|}tj ||} |\}}tj || z}tj || z}||ddtjfz||ddtjfzzS)Nr) npvstacklinalgqrTdiagnewaxisdotdetarctan2sincos) startendtbasisQRsignscsomegas \/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/scipy/spatial/_geometric_slerp.py_geometric_slerpr s Iucl # #E 9<< DAq q !A %E egaaam$$A egaaam$$A ucA aA Jq!  EJE3 q5yA q5yA 1QQQ ]# #cAaaam,<&< <<Hz>rz npt.ArrayLikerrtolreturnctj|tj}tj|tj}tj|}|jdkrt d|jdks |jdkrt d|j|jkrt d|jdks |jdkrt dtj||rtj|||jS||fD]F}tjtj |dd d st d Gt|tst d tj |}t||}tj|dd | rtjddtj|tj}|jd krtjd |jfS|jd kr5t%||tj|St%|||S)a 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. 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. .. versionchanged:: 1.17.0 Extrapolation is permitted, allowing values below `0` and above `1`. 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. 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: >>> import warnings >>> opposite_pole = np.array([-1, 0]) >>> with warnings.catch_warnings(): ... warnings.simplefilter("ignore", 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() 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() )dtyperz:The interpolation parameter value must be one dimensional.z1Start and end coordinates must be one-dimensionalz;The dimensions of start and end must match (have same size)rzLThe start and end coordinates must both be in at least two-dimensional spaceg?g& .>r)rtolatolz(start and end are not on a unit n-sphereztol must be a floatg@z_start and end are antipodes using the specified tolerance; this may cause ambiguous slerp paths) stacklevel)r asarrayfloat64ndim ValueErrorsize array_equallinspaceallcloser norm isinstancefloatfabsrwarningswarnemptyr atleast_1dravel)rrrr#coord coord_dists rrr!sH JuBJ / / /E *S + + +C 1 Avzz:;; ; zQ#(a--344 4 zSX;<< < zA~~A!"" 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