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diff --git a/libs/assimp/port/PyAssimp/scripts/transformations.py b/libs/assimp/port/PyAssimp/scripts/transformations.py deleted file mode 100644 index bf0cac9..0000000 --- a/libs/assimp/port/PyAssimp/scripts/transformations.py +++ /dev/null @@ -1,1705 +0,0 @@ -# -*- coding: utf-8 -*- -# transformations.py - -# Copyright (c) 2006, Christoph Gohlke -# Copyright (c) 2006-2009, The Regents of the University of California -# All rights reserved. -# -# Redistribution and use in source and binary forms, with or without -# modification, are permitted provided that the following conditions are met: -# -# * Redistributions of source code must retain the above copyright -# notice, this list of conditions and the following disclaimer. -# * Redistributions in binary form must reproduce the above copyright -# notice, this list of conditions and the following disclaimer in the -# documentation and/or other materials provided with the distribution. -# * Neither the name of the copyright holders nor the names of any -# contributors may be used to endorse or promote products derived -# from this software without specific prior written permission. -# -# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" -# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE -# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE -# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE -# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR -# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF -# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS -# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN -# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) -# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE -# POSSIBILITY OF SUCH DAMAGE. - -"""Homogeneous Transformation Matrices and Quaternions. - -A library for calculating 4x4 matrices for translating, rotating, reflecting, -scaling, shearing, projecting, orthogonalizing, and superimposing arrays of -3D homogeneous coordinates as well as for converting between rotation matrices, -Euler angles, and quaternions. Also includes an Arcball control object and -functions to decompose transformation matrices. - -:Authors: - `Christoph Gohlke <http://www.lfd.uci.edu/~gohlke/>`__, - Laboratory for Fluorescence Dynamics, University of California, Irvine - -:Version: 20090418 - -Requirements ------------- - -* `Python 2.6 <http://www.python.org>`__ -* `Numpy 1.3 <http://numpy.scipy.org>`__ -* `transformations.c 20090418 <http://www.lfd.uci.edu/~gohlke/>`__ - (optional implementation of some functions in C) - -Notes ------ - -Matrices (M) can be inverted using numpy.linalg.inv(M), concatenated using -numpy.dot(M0, M1), or used to transform homogeneous coordinates (v) using -numpy.dot(M, v) for shape (4, \*) "point of arrays", respectively -numpy.dot(v, M.T) for shape (\*, 4) "array of points". - -Calculations are carried out with numpy.float64 precision. - -This Python implementation is not optimized for speed. - -Vector, point, quaternion, and matrix function arguments are expected to be -"array like", i.e. tuple, list, or numpy arrays. - -Return types are numpy arrays unless specified otherwise. - -Angles are in radians unless specified otherwise. - -Quaternions ix+jy+kz+w are represented as [x, y, z, w]. - -Use the transpose of transformation matrices for OpenGL glMultMatrixd(). - -A triple of Euler angles can be applied/interpreted in 24 ways, which can -be specified using a 4 character string or encoded 4-tuple: - - *Axes 4-string*: e.g. 'sxyz' or 'ryxy' - - - first character : rotations are applied to 's'tatic or 'r'otating frame - - remaining characters : successive rotation axis 'x', 'y', or 'z' - - *Axes 4-tuple*: e.g. (0, 0, 0, 0) or (1, 1, 1, 1) - - - inner axis: code of axis ('x':0, 'y':1, 'z':2) of rightmost matrix. - - parity : even (0) if inner axis 'x' is followed by 'y', 'y' is followed - by 'z', or 'z' is followed by 'x'. Otherwise odd (1). - - repetition : first and last axis are same (1) or different (0). - - frame : rotations are applied to static (0) or rotating (1) frame. - -References ----------- - -(1) Matrices and transformations. Ronald Goldman. - In "Graphics Gems I", pp 472-475. Morgan Kaufmann, 1990. -(2) More matrices and transformations: shear and pseudo-perspective. - Ronald Goldman. In "Graphics Gems II", pp 320-323. Morgan Kaufmann, 1991. -(3) Decomposing a matrix into simple transformations. Spencer Thomas. - In "Graphics Gems II", pp 320-323. Morgan Kaufmann, 1991. -(4) Recovering the data from the transformation matrix. Ronald Goldman. - In "Graphics Gems II", pp 324-331. Morgan Kaufmann, 1991. -(5) Euler angle conversion. Ken Shoemake. - In "Graphics Gems IV", pp 222-229. Morgan Kaufmann, 1994. -(6) Arcball rotation control. Ken Shoemake. - In "Graphics Gems IV", pp 175-192. Morgan Kaufmann, 1994. -(7) Representing attitude: Euler angles, unit quaternions, and rotation - vectors. James Diebel. 2006. -(8) A discussion of the solution for the best rotation to relate two sets - of vectors. W Kabsch. Acta Cryst. 1978. A34, 827-828. -(9) Closed-form solution of absolute orientation using unit quaternions. - BKP Horn. J Opt Soc Am A. 1987. 4(4), 629-642. -(10) Quaternions. Ken Shoemake. - http://www.sfu.ca/~jwa3/cmpt461/files/quatut.pdf -(11) From quaternion to matrix and back. JMP van Waveren. 2005. - http://www.intel.com/cd/ids/developer/asmo-na/eng/293748.htm -(12) Uniform random rotations. Ken Shoemake. - In "Graphics Gems III", pp 124-132. Morgan Kaufmann, 1992. - - -Examples --------- - ->>> alpha, beta, gamma = 0.123, -1.234, 2.345 ->>> origin, xaxis, yaxis, zaxis = (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) ->>> I = identity_matrix() ->>> Rx = rotation_matrix(alpha, xaxis) ->>> Ry = rotation_matrix(beta, yaxis) ->>> Rz = rotation_matrix(gamma, zaxis) ->>> R = concatenate_matrices(Rx, Ry, Rz) ->>> euler = euler_from_matrix(R, 'rxyz') ->>> numpy.allclose([alpha, beta, gamma], euler) -True ->>> Re = euler_matrix(alpha, beta, gamma, 'rxyz') ->>> is_same_transform(R, Re) -True ->>> al, be, ga = euler_from_matrix(Re, 'rxyz') ->>> is_same_transform(Re, euler_matrix(al, be, ga, 'rxyz')) -True ->>> qx = quaternion_about_axis(alpha, xaxis) ->>> qy = quaternion_about_axis(beta, yaxis) ->>> qz = quaternion_about_axis(gamma, zaxis) ->>> q = quaternion_multiply(qx, qy) ->>> q = quaternion_multiply(q, qz) ->>> Rq = quaternion_matrix(q) ->>> is_same_transform(R, Rq) -True ->>> S = scale_matrix(1.23, origin) ->>> T = translation_matrix((1, 2, 3)) ->>> Z = shear_matrix(beta, xaxis, origin, zaxis) ->>> R = random_rotation_matrix(numpy.random.rand(3)) ->>> M = concatenate_matrices(T, R, Z, S) ->>> scale, shear, angles, trans, persp = decompose_matrix(M) ->>> numpy.allclose(scale, 1.23) -True ->>> numpy.allclose(trans, (1, 2, 3)) -True ->>> numpy.allclose(shear, (0, math.tan(beta), 0)) -True ->>> is_same_transform(R, euler_matrix(axes='sxyz', *angles)) -True ->>> M1 = compose_matrix(scale, shear, angles, trans, persp) ->>> is_same_transform(M, M1) -True - -""" - -from __future__ import division - -import warnings -import math - -import numpy - -# Documentation in HTML format can be generated with Epydoc -__docformat__ = "restructuredtext en" - - -def identity_matrix(): - """Return 4x4 identity/unit matrix. - - >>> I = identity_matrix() - >>> numpy.allclose(I, numpy.dot(I, I)) - True - >>> numpy.sum(I), numpy.trace(I) - (4.0, 4.0) - >>> numpy.allclose(I, numpy.identity(4, dtype=numpy.float64)) - True - - """ - return numpy.identity(4, dtype=numpy.float64) - - -def translation_matrix(direction): - """Return matrix to translate by direction vector. - - >>> v = numpy.random.random(3) - 0.5 - >>> numpy.allclose(v, translation_matrix(v)[:3, 3]) - True - - """ - M = numpy.identity(4) - M[:3, 3] = direction[:3] - return M - - -def translation_from_matrix(matrix): - """Return translation vector from translation matrix. - - >>> v0 = numpy.random.random(3) - 0.5 - >>> v1 = translation_from_matrix(translation_matrix(v0)) - >>> numpy.allclose(v0, v1) - True - - """ - return numpy.array(matrix, copy=False)[:3, 3].copy() - - -def reflection_matrix(point, normal): - """Return matrix to mirror at plane defined by point and normal vector. - - >>> v0 = numpy.random.random(4) - 0.5 - >>> v0[3] = 1.0 - >>> v1 = numpy.random.random(3) - 0.5 - >>> R = reflection_matrix(v0, v1) - >>> numpy.allclose(2., numpy.trace(R)) - True - >>> numpy.allclose(v0, numpy.dot(R, v0)) - True - >>> v2 = v0.copy() - >>> v2[:3] += v1 - >>> v3 = v0.copy() - >>> v2[:3] -= v1 - >>> numpy.allclose(v2, numpy.dot(R, v3)) - True - - """ - normal = unit_vector(normal[:3]) - M = numpy.identity(4) - M[:3, :3] -= 2.0 * numpy.outer(normal, normal) - M[:3, 3] = (2.0 * numpy.dot(point[:3], normal)) * normal - return M - - -def reflection_from_matrix(matrix): - """Return mirror plane point and normal vector from reflection matrix. - - >>> v0 = numpy.random.random(3) - 0.5 - >>> v1 = numpy.random.random(3) - 0.5 - >>> M0 = reflection_matrix(v0, v1) - >>> point, normal = reflection_from_matrix(M0) - >>> M1 = reflection_matrix(point, normal) - >>> is_same_transform(M0, M1) - True - - """ - M = numpy.array(matrix, dtype=numpy.float64, copy=False) - # normal: unit eigenvector corresponding to eigenvalue -1 - l, V = numpy.linalg.eig(M[:3, :3]) - i = numpy.where(abs(numpy.real(l) + 1.0) < 1e-8)[0] - if not len(i): - raise ValueError("no unit eigenvector corresponding to eigenvalue -1") - normal = numpy.real(V[:, i[0]]).squeeze() - # point: any unit eigenvector corresponding to eigenvalue 1 - l, V = numpy.linalg.eig(M) - i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0] - if not len(i): - raise ValueError("no unit eigenvector corresponding to eigenvalue 1") - point = numpy.real(V[:, i[-1]]).squeeze() - point /= point[3] - return point, normal - - -def rotation_matrix(angle, direction, point=None): - """Return matrix to rotate about axis defined by point and direction. - - >>> angle = (random.random() - 0.5) * (2*math.pi) - >>> direc = numpy.random.random(3) - 0.5 - >>> point = numpy.random.random(3) - 0.5 - >>> R0 = rotation_matrix(angle, direc, point) - >>> R1 = rotation_matrix(angle-2*math.pi, direc, point) - >>> is_same_transform(R0, R1) - True - >>> R0 = rotation_matrix(angle, direc, point) - >>> R1 = rotation_matrix(-angle, -direc, point) - >>> is_same_transform(R0, R1) - True - >>> I = numpy.identity(4, numpy.float64) - >>> numpy.allclose(I, rotation_matrix(math.pi*2, direc)) - True - >>> numpy.allclose(2., numpy.trace(rotation_matrix(math.pi/2, - ... direc, point))) - True - - """ - sina = math.sin(angle) - cosa = math.cos(angle) - direction = unit_vector(direction[:3]) - # rotation matrix around unit vector - R = numpy.array(((cosa, 0.0, 0.0), - (0.0, cosa, 0.0), - (0.0, 0.0, cosa)), dtype=numpy.float64) - R += numpy.outer(direction, direction) * (1.0 - cosa) - direction *= sina - R += numpy.array((( 0.0, -direction[2], direction[1]), - ( direction[2], 0.0, -direction[0]), - (-direction[1], direction[0], 0.0)), - dtype=numpy.float64) - M = numpy.identity(4) - M[:3, :3] = R - if point is not None: - # rotation not around origin - point = numpy.array(point[:3], dtype=numpy.float64, copy=False) - M[:3, 3] = point - numpy.dot(R, point) - return M - - -def rotation_from_matrix(matrix): - """Return rotation angle and axis from rotation matrix. - - >>> angle = (random.random() - 0.5) * (2*math.pi) - >>> direc = numpy.random.random(3) - 0.5 - >>> point = numpy.random.random(3) - 0.5 - >>> R0 = rotation_matrix(angle, direc, point) - >>> angle, direc, point = rotation_from_matrix(R0) - >>> R1 = rotation_matrix(angle, direc, point) - >>> is_same_transform(R0, R1) - True - - """ - R = numpy.array(matrix, dtype=numpy.float64, copy=False) - R33 = R[:3, :3] - # direction: unit eigenvector of R33 corresponding to eigenvalue of 1 - l, W = numpy.linalg.eig(R33.T) - i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0] - if not len(i): - raise ValueError("no unit eigenvector corresponding to eigenvalue 1") - direction = numpy.real(W[:, i[-1]]).squeeze() - # point: unit eigenvector of R33 corresponding to eigenvalue of 1 - l, Q = numpy.linalg.eig(R) - i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0] - if not len(i): - raise ValueError("no unit eigenvector corresponding to eigenvalue 1") - point = numpy.real(Q[:, i[-1]]).squeeze() - point /= point[3] - # rotation angle depending on direction - cosa = (numpy.trace(R33) - 1.0) / 2.0 - if abs(direction[2]) > 1e-8: - sina = (R[1, 0] + (cosa-1.0)*direction[0]*direction[1]) / direction[2] - elif abs(direction[1]) > 1e-8: - sina = (R[0, 2] + (cosa-1.0)*direction[0]*direction[2]) / direction[1] - else: - sina = (R[2, 1] + (cosa-1.0)*direction[1]*direction[2]) / direction[0] - angle = math.atan2(sina, cosa) - return angle, direction, point - - -def scale_matrix(factor, origin=None, direction=None): - """Return matrix to scale by factor around origin in direction. - - Use factor -1 for point symmetry. - - >>> v = (numpy.random.rand(4, 5) - 0.5) * 20.0 - >>> v[3] = 1.0 - >>> S = scale_matrix(-1.234) - >>> numpy.allclose(numpy.dot(S, v)[:3], -1.234*v[:3]) - True - >>> factor = random.random() * 10 - 5 - >>> origin = numpy.random.random(3) - 0.5 - >>> direct = numpy.random.random(3) - 0.5 - >>> S = scale_matrix(factor, origin) - >>> S = scale_matrix(factor, origin, direct) - - """ - if direction is None: - # uniform scaling - M = numpy.array(((factor, 0.0, 0.0, 0.0), - (0.0, factor, 0.0, 0.0), - (0.0, 0.0, factor, 0.0), - (0.0, 0.0, 0.0, 1.0)), dtype=numpy.float64) - if origin is not None: - M[:3, 3] = origin[:3] - M[:3, 3] *= 1.0 - factor - else: - # nonuniform scaling - direction = unit_vector(direction[:3]) - factor = 1.0 - factor - M = numpy.identity(4) - M[:3, :3] -= factor * numpy.outer(direction, direction) - if origin is not None: - M[:3, 3] = (factor * numpy.dot(origin[:3], direction)) * direction - return M - - -def scale_from_matrix(matrix): - """Return scaling factor, origin and direction from scaling matrix. - - >>> factor = random.random() * 10 - 5 - >>> origin = numpy.random.random(3) - 0.5 - >>> direct = numpy.random.random(3) - 0.5 - >>> S0 = scale_matrix(factor, origin) - >>> factor, origin, direction = scale_from_matrix(S0) - >>> S1 = scale_matrix(factor, origin, direction) - >>> is_same_transform(S0, S1) - True - >>> S0 = scale_matrix(factor, origin, direct) - >>> factor, origin, direction = scale_from_matrix(S0) - >>> S1 = scale_matrix(factor, origin, direction) - >>> is_same_transform(S0, S1) - True - - """ - M = numpy.array(matrix, dtype=numpy.float64, copy=False) - M33 = M[:3, :3] - factor = numpy.trace(M33) - 2.0 - try: - # direction: unit eigenvector corresponding to eigenvalue factor - l, V = numpy.linalg.eig(M33) - i = numpy.where(abs(numpy.real(l) - factor) < 1e-8)[0][0] - direction = numpy.real(V[:, i]).squeeze() - direction /= vector_norm(direction) - except IndexError: - # uniform scaling - factor = (factor + 2.0) / 3.0 - direction = None - # origin: any eigenvector corresponding to eigenvalue 1 - l, V = numpy.linalg.eig(M) - i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0] - if not len(i): - raise ValueError("no eigenvector corresponding to eigenvalue 1") - origin = numpy.real(V[:, i[-1]]).squeeze() - origin /= origin[3] - return factor, origin, direction - - -def projection_matrix(point, normal, direction=None, - perspective=None, pseudo=False): - """Return matrix to project onto plane defined by point and normal. - - Using either perspective point, projection direction, or none of both. - - If pseudo is True, perspective projections will preserve relative depth - such that Perspective = dot(Orthogonal, PseudoPerspective). - - >>> P = projection_matrix((0, 0, 0), (1, 0, 0)) - >>> numpy.allclose(P[1:, 1:], numpy.identity(4)[1:, 1:]) - True - >>> point = numpy.random.random(3) - 0.5 - >>> normal = numpy.random.random(3) - 0.5 - >>> direct = numpy.random.random(3) - 0.5 - >>> persp = numpy.random.random(3) - 0.5 - >>> P0 = projection_matrix(point, normal) - >>> P1 = projection_matrix(point, normal, direction=direct) - >>> P2 = projection_matrix(point, normal, perspective=persp) - >>> P3 = projection_matrix(point, normal, perspective=persp, pseudo=True) - >>> is_same_transform(P2, numpy.dot(P0, P3)) - True - >>> P = projection_matrix((3, 0, 0), (1, 1, 0), (1, 0, 0)) - >>> v0 = (numpy.random.rand(4, 5) - 0.5) * 20.0 - >>> v0[3] = 1.0 - >>> v1 = numpy.dot(P, v0) - >>> numpy.allclose(v1[1], v0[1]) - True - >>> numpy.allclose(v1[0], 3.0-v1[1]) - True - - """ - M = numpy.identity(4) - point = numpy.array(point[:3], dtype=numpy.float64, copy=False) - normal = unit_vector(normal[:3]) - if perspective is not None: - # perspective projection - perspective = numpy.array(perspective[:3], dtype=numpy.float64, - copy=False) - M[0, 0] = M[1, 1] = M[2, 2] = numpy.dot(perspective-point, normal) - M[:3, :3] -= numpy.outer(perspective, normal) - if pseudo: - # preserve relative depth - M[:3, :3] -= numpy.outer(normal, normal) - M[:3, 3] = numpy.dot(point, normal) * (perspective+normal) - else: - M[:3, 3] = numpy.dot(point, normal) * perspective - M[3, :3] = -normal - M[3, 3] = numpy.dot(perspective, normal) - elif direction is not None: - # parallel projection - direction = numpy.array(direction[:3], dtype=numpy.float64, copy=False) - scale = numpy.dot(direction, normal) - M[:3, :3] -= numpy.outer(direction, normal) / scale - M[:3, 3] = direction * (numpy.dot(point, normal) / scale) - else: - # orthogonal projection - M[:3, :3] -= numpy.outer(normal, normal) - M[:3, 3] = numpy.dot(point, normal) * normal - return M - - -def projection_from_matrix(matrix, pseudo=False): - """Return projection plane and perspective point from projection matrix. - - Return values are same as arguments for projection_matrix function: - point, normal, direction, perspective, and pseudo. - - >>> point = numpy.random.random(3) - 0.5 - >>> normal = numpy.random.random(3) - 0.5 - >>> direct = numpy.random.random(3) - 0.5 - >>> persp = numpy.random.random(3) - 0.5 - >>> P0 = projection_matrix(point, normal) - >>> result = projection_from_matrix(P0) - >>> P1 = projection_matrix(*result) - >>> is_same_transform(P0, P1) - True - >>> P0 = projection_matrix(point, normal, direct) - >>> result = projection_from_matrix(P0) - >>> P1 = projection_matrix(*result) - >>> is_same_transform(P0, P1) - True - >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=False) - >>> result = projection_from_matrix(P0, pseudo=False) - >>> P1 = projection_matrix(*result) - >>> is_same_transform(P0, P1) - True - >>> P0 = projection_matrix(point, normal, perspective=persp, pseudo=True) - >>> result = projection_from_matrix(P0, pseudo=True) - >>> P1 = projection_matrix(*result) - >>> is_same_transform(P0, P1) - True - - """ - M = numpy.array(matrix, dtype=numpy.float64, copy=False) - M33 = M[:3, :3] - l, V = numpy.linalg.eig(M) - i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0] - if not pseudo and len(i): - # point: any eigenvector corresponding to eigenvalue 1 - point = numpy.real(V[:, i[-1]]).squeeze() - point /= point[3] - # direction: unit eigenvector corresponding to eigenvalue 0 - l, V = numpy.linalg.eig(M33) - i = numpy.where(abs(numpy.real(l)) < 1e-8)[0] - if not len(i): - raise ValueError("no eigenvector corresponding to eigenvalue 0") - direction = numpy.real(V[:, i[0]]).squeeze() - direction /= vector_norm(direction) - # normal: unit eigenvector of M33.T corresponding to eigenvalue 0 - l, V = numpy.linalg.eig(M33.T) - i = numpy.where(abs(numpy.real(l)) < 1e-8)[0] - if len(i): - # parallel projection - normal = numpy.real(V[:, i[0]]).squeeze() - normal /= vector_norm(normal) - return point, normal, direction, None, False - else: - # orthogonal projection, where normal equals direction vector - return point, direction, None, None, False - else: - # perspective projection - i = numpy.where(abs(numpy.real(l)) > 1e-8)[0] - if not len(i): - raise ValueError( - "no eigenvector not corresponding to eigenvalue 0") - point = numpy.real(V[:, i[-1]]).squeeze() - point /= point[3] - normal = - M[3, :3] - perspective = M[:3, 3] / numpy.dot(point[:3], normal) - if pseudo: - perspective -= normal - return point, normal, None, perspective, pseudo - - -def clip_matrix(left, right, bottom, top, near, far, perspective=False): - """Return matrix to obtain normalized device coordinates from frustrum. - - The frustrum bounds are axis-aligned along x (left, right), - y (bottom, top) and z (near, far). - - Normalized device coordinates are in range [-1, 1] if coordinates are - inside the frustrum. - - If perspective is True the frustrum is a truncated pyramid with the - perspective point at origin and direction along z axis, otherwise an - orthographic canonical view volume (a box). - - Homogeneous coordinates transformed by the perspective clip matrix - need to be dehomogenized (divided by w coordinate). - - >>> frustrum = numpy.random.rand(6) - >>> frustrum[1] += frustrum[0] - >>> frustrum[3] += frustrum[2] - >>> frustrum[5] += frustrum[4] - >>> M = clip_matrix(*frustrum, perspective=False) - >>> numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1.0]) - array([-1., -1., -1., 1.]) - >>> numpy.dot(M, [frustrum[1], frustrum[3], frustrum[5], 1.0]) - array([ 1., 1., 1., 1.]) - >>> M = clip_matrix(*frustrum, perspective=True) - >>> v = numpy.dot(M, [frustrum[0], frustrum[2], frustrum[4], 1.0]) - >>> v / v[3] - array([-1., -1., -1., 1.]) - >>> v = numpy.dot(M, [frustrum[1], frustrum[3], frustrum[4], 1.0]) - >>> v / v[3] - array([ 1., 1., -1., 1.]) - - """ - if left >= right or bottom >= top or near >= far: - raise ValueError("invalid frustrum") - if perspective: - if near <= _EPS: - raise ValueError("invalid frustrum: near <= 0") - t = 2.0 * near - M = ((-t/(right-left), 0.0, (right+left)/(right-left), 0.0), - (0.0, -t/(top-bottom), (top+bottom)/(top-bottom), 0.0), - (0.0, 0.0, -(far+near)/(far-near), t*far/(far-near)), - (0.0, 0.0, -1.0, 0.0)) - else: - M = ((2.0/(right-left), 0.0, 0.0, (right+left)/(left-right)), - (0.0, 2.0/(top-bottom), 0.0, (top+bottom)/(bottom-top)), - (0.0, 0.0, 2.0/(far-near), (far+near)/(near-far)), - (0.0, 0.0, 0.0, 1.0)) - return numpy.array(M, dtype=numpy.float64) - - -def shear_matrix(angle, direction, point, normal): - """Return matrix to shear by angle along direction vector on shear plane. - - The shear plane is defined by a point and normal vector. The direction - vector must be orthogonal to the plane's normal vector. - - A point P is transformed by the shear matrix into P" such that - the vector P-P" is parallel to the direction vector and its extent is - given by the angle of P-P'-P", where P' is the orthogonal projection - of P onto the shear plane. - - >>> angle = (random.random() - 0.5) * 4*math.pi - >>> direct = numpy.random.random(3) - 0.5 - >>> point = numpy.random.random(3) - 0.5 - >>> normal = numpy.cross(direct, numpy.random.random(3)) - >>> S = shear_matrix(angle, direct, point, normal) - >>> numpy.allclose(1.0, numpy.linalg.det(S)) - True - - """ - normal = unit_vector(normal[:3]) - direction = unit_vector(direction[:3]) - if abs(numpy.dot(normal, direction)) > 1e-6: - raise ValueError("direction and normal vectors are not orthogonal") - angle = math.tan(angle) - M = numpy.identity(4) - M[:3, :3] += angle * numpy.outer(direction, normal) - M[:3, 3] = -angle * numpy.dot(point[:3], normal) * direction - return M - - -def shear_from_matrix(matrix): - """Return shear angle, direction and plane from shear matrix. - - >>> angle = (random.random() - 0.5) * 4*math.pi - >>> direct = numpy.random.random(3) - 0.5 - >>> point = numpy.random.random(3) - 0.5 - >>> normal = numpy.cross(direct, numpy.random.random(3)) - >>> S0 = shear_matrix(angle, direct, point, normal) - >>> angle, direct, point, normal = shear_from_matrix(S0) - >>> S1 = shear_matrix(angle, direct, point, normal) - >>> is_same_transform(S0, S1) - True - - """ - M = numpy.array(matrix, dtype=numpy.float64, copy=False) - M33 = M[:3, :3] - # normal: cross independent eigenvectors corresponding to the eigenvalue 1 - l, V = numpy.linalg.eig(M33) - i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-4)[0] - if len(i) < 2: - raise ValueError("No two linear independent eigenvectors found %s" % l) - V = numpy.real(V[:, i]).squeeze().T - lenorm = -1.0 - for i0, i1 in ((0, 1), (0, 2), (1, 2)): - n = numpy.cross(V[i0], V[i1]) - l = vector_norm(n) - if l > lenorm: - lenorm = l - normal = n - normal /= lenorm - # direction and angle - direction = numpy.dot(M33 - numpy.identity(3), normal) - angle = vector_norm(direction) - direction /= angle - angle = math.atan(angle) - # point: eigenvector corresponding to eigenvalue 1 - l, V = numpy.linalg.eig(M) - i = numpy.where(abs(numpy.real(l) - 1.0) < 1e-8)[0] - if not len(i): - raise ValueError("no eigenvector corresponding to eigenvalue 1") - point = numpy.real(V[:, i[-1]]).squeeze() - point /= point[3] - return angle, direction, point, normal - - -def decompose_matrix(matrix): - """Return sequence of transformations from transformation matrix. - - matrix : array_like - Non-degenerative homogeneous transformation matrix - - Return tuple of: - scale : vector of 3 scaling factors - shear : list of shear factors for x-y, x-z, y-z axes - angles : list of Euler angles about static x, y, z axes - translate : translation vector along x, y, z axes - perspective : perspective partition of matrix - - Raise ValueError if matrix is of wrong type or degenerative. - - >>> T0 = translation_matrix((1, 2, 3)) - >>> scale, shear, angles, trans, persp = decompose_matrix(T0) - >>> T1 = translation_matrix(trans) - >>> numpy.allclose(T0, T1) - True - >>> S = scale_matrix(0.123) - >>> scale, shear, angles, trans, persp = decompose_matrix(S) - >>> scale[0] - 0.123 - >>> R0 = euler_matrix(1, 2, 3) - >>> scale, shear, angles, trans, persp = decompose_matrix(R0) - >>> R1 = euler_matrix(*angles) - >>> numpy.allclose(R0, R1) - True - - """ - M = numpy.array(matrix, dtype=numpy.float64, copy=True).T - if abs(M[3, 3]) < _EPS: - raise ValueError("M[3, 3] is zero") - M /= M[3, 3] - P = M.copy() - P[:, 3] = 0, 0, 0, 1 - if not numpy.linalg.det(P): - raise ValueError("Matrix is singular") - - scale = numpy.zeros((3, ), dtype=numpy.float64) - shear = [0, 0, 0] - angles = [0, 0, 0] - - if any(abs(M[:3, 3]) > _EPS): - perspective = numpy.dot(M[:, 3], numpy.linalg.inv(P.T)) - M[:, 3] = 0, 0, 0, 1 - else: - perspective = numpy.array((0, 0, 0, 1), dtype=numpy.float64) - - translate = M[3, :3].copy() - M[3, :3] = 0 - - row = M[:3, :3].copy() - scale[0] = vector_norm(row[0]) - row[0] /= scale[0] - shear[0] = numpy.dot(row[0], row[1]) - row[1] -= row[0] * shear[0] - scale[1] = vector_norm(row[1]) - row[1] /= scale[1] - shear[0] /= scale[1] - shear[1] = numpy.dot(row[0], row[2]) - row[2] -= row[0] * shear[1] - shear[2] = numpy.dot(row[1], row[2]) - row[2] -= row[1] * shear[2] - scale[2] = vector_norm(row[2]) - row[2] /= scale[2] - shear[1:] /= scale[2] - - if numpy.dot(row[0], numpy.cross(row[1], row[2])) < 0: - scale *= -1 - row *= -1 - - angles[1] = math.asin(-row[0, 2]) - if math.cos(angles[1]): - angles[0] = math.atan2(row[1, 2], row[2, 2]) - angles[2] = math.atan2(row[0, 1], row[0, 0]) - else: - #angles[0] = math.atan2(row[1, 0], row[1, 1]) - angles[0] = math.atan2(-row[2, 1], row[1, 1]) - angles[2] = 0.0 - - return scale, shear, angles, translate, perspective - - -def compose_matrix(scale=None, shear=None, angles=None, translate=None, - perspective=None): - """Return transformation matrix from sequence of transformations. - - This is the inverse of the decompose_matrix function. - - Sequence of transformations: - scale : vector of 3 scaling factors - shear : list of shear factors for x-y, x-z, y-z axes - angles : list of Euler angles about static x, y, z axes - translate : translation vector along x, y, z axes - perspective : perspective partition of matrix - - >>> scale = numpy.random.random(3) - 0.5 - >>> shear = numpy.random.random(3) - 0.5 - >>> angles = (numpy.random.random(3) - 0.5) * (2*math.pi) - >>> trans = numpy.random.random(3) - 0.5 - >>> persp = numpy.random.random(4) - 0.5 - >>> M0 = compose_matrix(scale, shear, angles, trans, persp) - >>> result = decompose_matrix(M0) - >>> M1 = compose_matrix(*result) - >>> is_same_transform(M0, M1) - True - - """ - M = numpy.identity(4) - if perspective is not None: - P = numpy.identity(4) - P[3, :] = perspective[:4] - M = numpy.dot(M, P) - if translate is not None: - T = numpy.identity(4) - T[:3, 3] = translate[:3] - M = numpy.dot(M, T) - if angles is not None: - R = euler_matrix(angles[0], angles[1], angles[2], 'sxyz') - M = numpy.dot(M, R) - if shear is not None: - Z = numpy.identity(4) - Z[1, 2] = shear[2] - Z[0, 2] = shear[1] - Z[0, 1] = shear[0] - M = numpy.dot(M, Z) - if scale is not None: - S = numpy.identity(4) - S[0, 0] = scale[0] - S[1, 1] = scale[1] - S[2, 2] = scale[2] - M = numpy.dot(M, S) - M /= M[3, 3] - return M - - -def orthogonalization_matrix(lengths, angles): - """Return orthogonalization matrix for crystallographic cell coordinates. - - Angles are expected in degrees. - - The de-orthogonalization matrix is the inverse. - - >>> O = orthogonalization_matrix((10., 10., 10.), (90., 90., 90.)) - >>> numpy.allclose(O[:3, :3], numpy.identity(3, float) * 10) - True - >>> O = orthogonalization_matrix([9.8, 12.0, 15.5], [87.2, 80.7, 69.7]) - >>> numpy.allclose(numpy.sum(O), 43.063229) - True - - """ - a, b, c = lengths - angles = numpy.radians(angles) - sina, sinb, _ = numpy.sin(angles) - cosa, cosb, cosg = numpy.cos(angles) - co = (cosa * cosb - cosg) / (sina * sinb) - return numpy.array(( - ( a*sinb*math.sqrt(1.0-co*co), 0.0, 0.0, 0.0), - (-a*sinb*co, b*sina, 0.0, 0.0), - ( a*cosb, b*cosa, c, 0.0), - ( 0.0, 0.0, 0.0, 1.0)), - dtype=numpy.float64) - - -def superimposition_matrix(v0, v1, scaling=False, usesvd=True): - """Return matrix to transform given vector set into second vector set. - - v0 and v1 are shape (3, \*) or (4, \*) arrays of at least 3 vectors. - - If usesvd is True, the weighted sum of squared deviations (RMSD) is - minimized according to the algorithm by W. Kabsch [8]. Otherwise the - quaternion based algorithm by B. Horn [9] is used (slower when using - this Python implementation). - - The returned matrix performs rotation, translation and uniform scaling - (if specified). - - >>> v0 = numpy.random.rand(3, 10) - >>> M = superimposition_matrix(v0, v0) - >>> numpy.allclose(M, numpy.identity(4)) - True - >>> R = random_rotation_matrix(numpy.random.random(3)) - >>> v0 = ((1,0,0), (0,1,0), (0,0,1), (1,1,1)) - >>> v1 = numpy.dot(R, v0) - >>> M = superimposition_matrix(v0, v1) - >>> numpy.allclose(v1, numpy.dot(M, v0)) - True - >>> v0 = (numpy.random.rand(4, 100) - 0.5) * 20.0 - >>> v0[3] = 1.0 - >>> v1 = numpy.dot(R, v0) - >>> M = superimposition_matrix(v0, v1) - >>> numpy.allclose(v1, numpy.dot(M, v0)) - True - >>> S = scale_matrix(random.random()) - >>> T = translation_matrix(numpy.random.random(3)-0.5) - >>> M = concatenate_matrices(T, R, S) - >>> v1 = numpy.dot(M, v0) - >>> v0[:3] += numpy.random.normal(0.0, 1e-9, 300).reshape(3, -1) - >>> M = superimposition_matrix(v0, v1, scaling=True) - >>> numpy.allclose(v1, numpy.dot(M, v0)) - True - >>> M = superimposition_matrix(v0, v1, scaling=True, usesvd=False) - >>> numpy.allclose(v1, numpy.dot(M, v0)) - True - >>> v = numpy.empty((4, 100, 3), dtype=numpy.float64) - >>> v[:, :, 0] = v0 - >>> M = superimposition_matrix(v0, v1, scaling=True, usesvd=False) - >>> numpy.allclose(v1, numpy.dot(M, v[:, :, 0])) - True - - """ - v0 = numpy.array(v0, dtype=numpy.float64, copy=False)[:3] - v1 = numpy.array(v1, dtype=numpy.float64, copy=False)[:3] - - if v0.shape != v1.shape or v0.shape[1] < 3: - raise ValueError("Vector sets are of wrong shape or type.") - - # move centroids to origin - t0 = numpy.mean(v0, axis=1) - t1 = numpy.mean(v1, axis=1) - v0 = v0 - t0.reshape(3, 1) - v1 = v1 - t1.reshape(3, 1) - - if usesvd: - # Singular Value Decomposition of covariance matrix - u, s, vh = numpy.linalg.svd(numpy.dot(v1, v0.T)) - # rotation matrix from SVD orthonormal bases - R = numpy.dot(u, vh) - if numpy.linalg.det(R) < 0.0: - # R does not constitute right handed system - R -= numpy.outer(u[:, 2], vh[2, :]*2.0) - s[-1] *= -1.0 - # homogeneous transformation matrix - M = numpy.identity(4) - M[:3, :3] = R - else: - # compute symmetric matrix N - xx, yy, zz = numpy.sum(v0 * v1, axis=1) - xy, yz, zx = numpy.sum(v0 * numpy.roll(v1, -1, axis=0), axis=1) - xz, yx, zy = numpy.sum(v0 * numpy.roll(v1, -2, axis=0), axis=1) - N = ((xx+yy+zz, yz-zy, zx-xz, xy-yx), - (yz-zy, xx-yy-zz, xy+yx, zx+xz), - (zx-xz, xy+yx, -xx+yy-zz, yz+zy), - (xy-yx, zx+xz, yz+zy, -xx-yy+zz)) - # quaternion: eigenvector corresponding to most positive eigenvalue - l, V = numpy.linalg.eig(N) - q = V[:, numpy.argmax(l)] - q /= vector_norm(q) # unit quaternion - q = numpy.roll(q, -1) # move w component to end - # homogeneous transformation matrix - M = quaternion_matrix(q) - - # scale: ratio of rms deviations from centroid - if scaling: - v0 *= v0 - v1 *= v1 - M[:3, :3] *= math.sqrt(numpy.sum(v1) / numpy.sum(v0)) - - # translation - M[:3, 3] = t1 - T = numpy.identity(4) - T[:3, 3] = -t0 - M = numpy.dot(M, T) - return M - - -def euler_matrix(ai, aj, ak, axes='sxyz'): - """Return homogeneous rotation matrix from Euler angles and axis sequence. - - ai, aj, ak : Euler's roll, pitch and yaw angles - axes : One of 24 axis sequences as string or encoded tuple - - >>> R = euler_matrix(1, 2, 3, 'syxz') - >>> numpy.allclose(numpy.sum(R[0]), -1.34786452) - True - >>> R = euler_matrix(1, 2, 3, (0, 1, 0, 1)) - >>> numpy.allclose(numpy.sum(R[0]), -0.383436184) - True - >>> ai, aj, ak = (4.0*math.pi) * (numpy.random.random(3) - 0.5) - >>> for axes in _AXES2TUPLE.keys(): - ... R = euler_matrix(ai, aj, ak, axes) - >>> for axes in _TUPLE2AXES.keys(): - ... R = euler_matrix(ai, aj, ak, axes) - - """ - try: - firstaxis, parity, repetition, frame = _AXES2TUPLE[axes] - except (AttributeError, KeyError): - _ = _TUPLE2AXES[axes] - firstaxis, parity, repetition, frame = axes - - i = firstaxis - j = _NEXT_AXIS[i+parity] - k = _NEXT_AXIS[i-parity+1] - - if frame: - ai, ak = ak, ai - if parity: - ai, aj, ak = -ai, -aj, -ak - - si, sj, sk = math.sin(ai), math.sin(aj), math.sin(ak) - ci, cj, ck = math.cos(ai), math.cos(aj), math.cos(ak) - cc, cs = ci*ck, ci*sk - sc, ss = si*ck, si*sk - - M = numpy.identity(4) - if repetition: - M[i, i] = cj - M[i, j] = sj*si - M[i, k] = sj*ci - M[j, i] = sj*sk - M[j, j] = -cj*ss+cc - M[j, k] = -cj*cs-sc - M[k, i] = -sj*ck - M[k, j] = cj*sc+cs - M[k, k] = cj*cc-ss - else: - M[i, i] = cj*ck - M[i, j] = sj*sc-cs - M[i, k] = sj*cc+ss - M[j, i] = cj*sk - M[j, j] = sj*ss+cc - M[j, k] = sj*cs-sc - M[k, i] = -sj - M[k, j] = cj*si - M[k, k] = cj*ci - return M - - -def euler_from_matrix(matrix, axes='sxyz'): - """Return Euler angles from rotation matrix for specified axis sequence. - - axes : One of 24 axis sequences as string or encoded tuple - - Note that many Euler angle triplets can describe one matrix. - - >>> R0 = euler_matrix(1, 2, 3, 'syxz') - >>> al, be, ga = euler_from_matrix(R0, 'syxz') - >>> R1 = euler_matrix(al, be, ga, 'syxz') - >>> numpy.allclose(R0, R1) - True - >>> angles = (4.0*math.pi) * (numpy.random.random(3) - 0.5) - >>> for axes in _AXES2TUPLE.keys(): - ... R0 = euler_matrix(axes=axes, *angles) - ... R1 = euler_matrix(axes=axes, *euler_from_matrix(R0, axes)) - ... if not numpy.allclose(R0, R1): print axes, "failed" - - """ - try: - firstaxis, parity, repetition, frame = _AXES2TUPLE[axes.lower()] - except (AttributeError, KeyError): - _ = _TUPLE2AXES[axes] - firstaxis, parity, repetition, frame = axes - - i = firstaxis - j = _NEXT_AXIS[i+parity] - k = _NEXT_AXIS[i-parity+1] - - M = numpy.array(matrix, dtype=numpy.float64, copy=False)[:3, :3] - if repetition: - sy = math.sqrt(M[i, j]*M[i, j] + M[i, k]*M[i, k]) - if sy > _EPS: - ax = math.atan2( M[i, j], M[i, k]) - ay = math.atan2( sy, M[i, i]) - az = math.atan2( M[j, i], -M[k, i]) - else: - ax = math.atan2(-M[j, k], M[j, j]) - ay = math.atan2( sy, M[i, i]) - az = 0.0 - else: - cy = math.sqrt(M[i, i]*M[i, i] + M[j, i]*M[j, i]) - if cy > _EPS: - ax = math.atan2( M[k, j], M[k, k]) - ay = math.atan2(-M[k, i], cy) - az = math.atan2( M[j, i], M[i, i]) - else: - ax = math.atan2(-M[j, k], M[j, j]) - ay = math.atan2(-M[k, i], cy) - az = 0.0 - - if parity: - ax, ay, az = -ax, -ay, -az - if frame: - ax, az = az, ax - return ax, ay, az - - -def euler_from_quaternion(quaternion, axes='sxyz'): - """Return Euler angles from quaternion for specified axis sequence. - - >>> angles = euler_from_quaternion([0.06146124, 0, 0, 0.99810947]) - >>> numpy.allclose(angles, [0.123, 0, 0]) - True - - """ - return euler_from_matrix(quaternion_matrix(quaternion), axes) - - -def quaternion_from_euler(ai, aj, ak, axes='sxyz'): - """Return quaternion from Euler angles and axis sequence. - - ai, aj, ak : Euler's roll, pitch and yaw angles - axes : One of 24 axis sequences as string or encoded tuple - - >>> q = quaternion_from_euler(1, 2, 3, 'ryxz') - >>> numpy.allclose(q, [0.310622, -0.718287, 0.444435, 0.435953]) - True - - """ - try: - firstaxis, parity, repetition, frame = _AXES2TUPLE[axes.lower()] - except (AttributeError, KeyError): - _ = _TUPLE2AXES[axes] - firstaxis, parity, repetition, frame = axes - - i = firstaxis - j = _NEXT_AXIS[i+parity] - k = _NEXT_AXIS[i-parity+1] - - if frame: - ai, ak = ak, ai - if parity: - aj = -aj - - ai /= 2.0 - aj /= 2.0 - ak /= 2.0 - ci = math.cos(ai) - si = math.sin(ai) - cj = math.cos(aj) - sj = math.sin(aj) - ck = math.cos(ak) - sk = math.sin(ak) - cc = ci*ck - cs = ci*sk - sc = si*ck - ss = si*sk - - quaternion = numpy.empty((4, ), dtype=numpy.float64) - if repetition: - quaternion[i] = cj*(cs + sc) - quaternion[j] = sj*(cc + ss) - quaternion[k] = sj*(cs - sc) - quaternion[3] = cj*(cc - ss) - else: - quaternion[i] = cj*sc - sj*cs - quaternion[j] = cj*ss + sj*cc - quaternion[k] = cj*cs - sj*sc - quaternion[3] = cj*cc + sj*ss - if parity: - quaternion[j] *= -1 - - return quaternion - - -def quaternion_about_axis(angle, axis): - """Return quaternion for rotation about axis. - - >>> q = quaternion_about_axis(0.123, (1, 0, 0)) - >>> numpy.allclose(q, [0.06146124, 0, 0, 0.99810947]) - True - - """ - quaternion = numpy.zeros((4, ), dtype=numpy.float64) - quaternion[:3] = axis[:3] - qlen = vector_norm(quaternion) - if qlen > _EPS: - quaternion *= math.sin(angle/2.0) / qlen - quaternion[3] = math.cos(angle/2.0) - return quaternion - - -def quaternion_matrix(quaternion): - """Return homogeneous rotation matrix from quaternion. - - >>> R = quaternion_matrix([0.06146124, 0, 0, 0.99810947]) - >>> numpy.allclose(R, rotation_matrix(0.123, (1, 0, 0))) - True - - """ - q = numpy.array(quaternion[:4], dtype=numpy.float64, copy=True) - nq = numpy.dot(q, q) - if nq < _EPS: - return numpy.identity(4) - q *= math.sqrt(2.0 / nq) - q = numpy.outer(q, q) - return numpy.array(( - (1.0-q[1, 1]-q[2, 2], q[0, 1]-q[2, 3], q[0, 2]+q[1, 3], 0.0), - ( q[0, 1]+q[2, 3], 1.0-q[0, 0]-q[2, 2], q[1, 2]-q[0, 3], 0.0), - ( q[0, 2]-q[1, 3], q[1, 2]+q[0, 3], 1.0-q[0, 0]-q[1, 1], 0.0), - ( 0.0, 0.0, 0.0, 1.0) - ), dtype=numpy.float64) - - -def quaternion_from_matrix(matrix): - """Return quaternion from rotation matrix. - - >>> R = rotation_matrix(0.123, (1, 2, 3)) - >>> q = quaternion_from_matrix(R) - >>> numpy.allclose(q, [0.0164262, 0.0328524, 0.0492786, 0.9981095]) - True - - """ - q = numpy.empty((4, ), dtype=numpy.float64) - M = numpy.array(matrix, dtype=numpy.float64, copy=False)[:4, :4] - t = numpy.trace(M) - if t > M[3, 3]: - q[3] = t - q[2] = M[1, 0] - M[0, 1] - q[1] = M[0, 2] - M[2, 0] - q[0] = M[2, 1] - M[1, 2] - else: - i, j, k = 0, 1, 2 - if M[1, 1] > M[0, 0]: - i, j, k = 1, 2, 0 - if M[2, 2] > M[i, i]: - i, j, k = 2, 0, 1 - t = M[i, i] - (M[j, j] + M[k, k]) + M[3, 3] - q[i] = t - q[j] = M[i, j] + M[j, i] - q[k] = M[k, i] + M[i, k] - q[3] = M[k, j] - M[j, k] - q *= 0.5 / math.sqrt(t * M[3, 3]) - return q - - -def quaternion_multiply(quaternion1, quaternion0): - """Return multiplication of two quaternions. - - >>> q = quaternion_multiply([1, -2, 3, 4], [-5, 6, 7, 8]) - >>> numpy.allclose(q, [-44, -14, 48, 28]) - True - - """ - x0, y0, z0, w0 = quaternion0 - x1, y1, z1, w1 = quaternion1 - return numpy.array(( - x1*w0 + y1*z0 - z1*y0 + w1*x0, - -x1*z0 + y1*w0 + z1*x0 + w1*y0, - x1*y0 - y1*x0 + z1*w0 + w1*z0, - -x1*x0 - y1*y0 - z1*z0 + w1*w0), dtype=numpy.float64) - - -def quaternion_conjugate(quaternion): - """Return conjugate of quaternion. - - >>> q0 = random_quaternion() - >>> q1 = quaternion_conjugate(q0) - >>> q1[3] == q0[3] and all(q1[:3] == -q0[:3]) - True - - """ - return numpy.array((-quaternion[0], -quaternion[1], - -quaternion[2], quaternion[3]), dtype=numpy.float64) - - -def quaternion_inverse(quaternion): - """Return inverse of quaternion. - - >>> q0 = random_quaternion() - >>> q1 = quaternion_inverse(q0) - >>> numpy.allclose(quaternion_multiply(q0, q1), [0, 0, 0, 1]) - True - - """ - return quaternion_conjugate(quaternion) / numpy.dot(quaternion, quaternion) - - -def quaternion_slerp(quat0, quat1, fraction, spin=0, shortestpath=True): - """Return spherical linear interpolation between two quaternions. - - >>> q0 = random_quaternion() - >>> q1 = random_quaternion() - >>> q = quaternion_slerp(q0, q1, 0.0) - >>> numpy.allclose(q, q0) - True - >>> q = quaternion_slerp(q0, q1, 1.0, 1) - >>> numpy.allclose(q, q1) - True - >>> q = quaternion_slerp(q0, q1, 0.5) - >>> angle = math.acos(numpy.dot(q0, q)) - >>> numpy.allclose(2.0, math.acos(numpy.dot(q0, q1)) / angle) or \ - numpy.allclose(2.0, math.acos(-numpy.dot(q0, q1)) / angle) - True - - """ - q0 = unit_vector(quat0[:4]) - q1 = unit_vector(quat1[:4]) - if fraction == 0.0: - return q0 - elif fraction == 1.0: - return q1 - d = numpy.dot(q0, q1) - if abs(abs(d) - 1.0) < _EPS: - return q0 - if shortestpath and d < 0.0: - # invert rotation - d = -d - q1 *= -1.0 - angle = math.acos(d) + spin * math.pi - if abs(angle) < _EPS: - return q0 - isin = 1.0 / math.sin(angle) - q0 *= math.sin((1.0 - fraction) * angle) * isin - q1 *= math.sin(fraction * angle) * isin - q0 += q1 - return q0 - - -def random_quaternion(rand=None): - """Return uniform random unit quaternion. - - rand: array like or None - Three independent random variables that are uniformly distributed - between 0 and 1. - - >>> q = random_quaternion() - >>> numpy.allclose(1.0, vector_norm(q)) - True - >>> q = random_quaternion(numpy.random.random(3)) - >>> q.shape - (4,) - - """ - if rand is None: - rand = numpy.random.rand(3) - else: - assert len(rand) == 3 - r1 = numpy.sqrt(1.0 - rand[0]) - r2 = numpy.sqrt(rand[0]) - pi2 = math.pi * 2.0 - t1 = pi2 * rand[1] - t2 = pi2 * rand[2] - return numpy.array((numpy.sin(t1)*r1, - numpy.cos(t1)*r1, - numpy.sin(t2)*r2, - numpy.cos(t2)*r2), dtype=numpy.float64) - - -def random_rotation_matrix(rand=None): - """Return uniform random rotation matrix. - - rnd: array like - Three independent random variables that are uniformly distributed - between 0 and 1 for each returned quaternion. - - >>> R = random_rotation_matrix() - >>> numpy.allclose(numpy.dot(R.T, R), numpy.identity(4)) - True - - """ - return quaternion_matrix(random_quaternion(rand)) - - -class Arcball(object): - """Virtual Trackball Control. - - >>> ball = Arcball() - >>> ball = Arcball(initial=numpy.identity(4)) - >>> ball.place([320, 320], 320) - >>> ball.down([500, 250]) - >>> ball.drag([475, 275]) - >>> R = ball.matrix() - >>> numpy.allclose(numpy.sum(R), 3.90583455) - True - >>> ball = Arcball(initial=[0, 0, 0, 1]) - >>> ball.place([320, 320], 320) - >>> ball.setaxes([1,1,0], [-1, 1, 0]) - >>> ball.setconstrain(True) - >>> ball.down([400, 200]) - >>> ball.drag([200, 400]) - >>> R = ball.matrix() - >>> numpy.allclose(numpy.sum(R), 0.2055924) - True - >>> ball.next() - - """ - - def __init__(self, initial=None): - """Initialize virtual trackball control. - - initial : quaternion or rotation matrix - - """ - self._axis = None - self._axes = None - self._radius = 1.0 - self._center = [0.0, 0.0] - self._vdown = numpy.array([0, 0, 1], dtype=numpy.float64) - self._constrain = False - - if initial is None: - self._qdown = numpy.array([0, 0, 0, 1], dtype=numpy.float64) - else: - initial = numpy.array(initial, dtype=numpy.float64) - if initial.shape == (4, 4): - self._qdown = quaternion_from_matrix(initial) - elif initial.shape == (4, ): - initial /= vector_norm(initial) - self._qdown = initial - else: - raise ValueError("initial not a quaternion or matrix.") - - self._qnow = self._qpre = self._qdown - - def place(self, center, radius): - """Place Arcball, e.g. when window size changes. - - center : sequence[2] - Window coordinates of trackball center. - radius : float - Radius of trackball in window coordinates. - - """ - self._radius = float(radius) - self._center[0] = center[0] - self._center[1] = center[1] - - def setaxes(self, *axes): - """Set axes to constrain rotations.""" - if axes is None: - self._axes = None - else: - self._axes = [unit_vector(axis) for axis in axes] - - def setconstrain(self, constrain): - """Set state of constrain to axis mode.""" - self._constrain = constrain == True - - def getconstrain(self): - """Return state of constrain to axis mode.""" - return self._constrain - - def down(self, point): - """Set initial cursor window coordinates and pick constrain-axis.""" - self._vdown = arcball_map_to_sphere(point, self._center, self._radius) - self._qdown = self._qpre = self._qnow - - if self._constrain and self._axes is not None: - self._axis = arcball_nearest_axis(self._vdown, self._axes) - self._vdown = arcball_constrain_to_axis(self._vdown, self._axis) - else: - self._axis = None - - def drag(self, point): - """Update current cursor window coordinates.""" - vnow = arcball_map_to_sphere(point, self._center, self._radius) - - if self._axis is not None: - vnow = arcball_constrain_to_axis(vnow, self._axis) - - self._qpre = self._qnow - - t = numpy.cross(self._vdown, vnow) - if numpy.dot(t, t) < _EPS: - self._qnow = self._qdown - else: - q = [t[0], t[1], t[2], numpy.dot(self._vdown, vnow)] - self._qnow = quaternion_multiply(q, self._qdown) - - def next(self, acceleration=0.0): - """Continue rotation in direction of last drag.""" - q = quaternion_slerp(self._qpre, self._qnow, 2.0+acceleration, False) - self._qpre, self._qnow = self._qnow, q - - def matrix(self): - """Return homogeneous rotation matrix.""" - return quaternion_matrix(self._qnow) - - -def arcball_map_to_sphere(point, center, radius): - """Return unit sphere coordinates from window coordinates.""" - v = numpy.array(((point[0] - center[0]) / radius, - (center[1] - point[1]) / radius, - 0.0), dtype=numpy.float64) - n = v[0]*v[0] + v[1]*v[1] - if n > 1.0: - v /= math.sqrt(n) # position outside of sphere - else: - v[2] = math.sqrt(1.0 - n) - return v - - -def arcball_constrain_to_axis(point, axis): - """Return sphere point perpendicular to axis.""" - v = numpy.array(point, dtype=numpy.float64, copy=True) - a = numpy.array(axis, dtype=numpy.float64, copy=True) - v -= a * numpy.dot(a, v) # on plane - n = vector_norm(v) - if n > _EPS: - if v[2] < 0.0: - v *= -1.0 - v /= n - return v - if a[2] == 1.0: - return numpy.array([1, 0, 0], dtype=numpy.float64) - return unit_vector([-a[1], a[0], 0]) - - -def arcball_nearest_axis(point, axes): - """Return axis, which arc is nearest to point.""" - point = numpy.array(point, dtype=numpy.float64, copy=False) - nearest = None - mx = -1.0 - for axis in axes: - t = numpy.dot(arcball_constrain_to_axis(point, axis), point) - if t > mx: - nearest = axis - mx = t - return nearest - - -# epsilon for testing whether a number is close to zero -_EPS = numpy.finfo(float).eps * 4.0 - -# axis sequences for Euler angles -_NEXT_AXIS = [1, 2, 0, 1] - -# map axes strings to/from tuples of inner axis, parity, repetition, frame -_AXES2TUPLE = { - 'sxyz': (0, 0, 0, 0), 'sxyx': (0, 0, 1, 0), 'sxzy': (0, 1, 0, 0), - 'sxzx': (0, 1, 1, 0), 'syzx': (1, 0, 0, 0), 'syzy': (1, 0, 1, 0), - 'syxz': (1, 1, 0, 0), 'syxy': (1, 1, 1, 0), 'szxy': (2, 0, 0, 0), - 'szxz': (2, 0, 1, 0), 'szyx': (2, 1, 0, 0), 'szyz': (2, 1, 1, 0), - 'rzyx': (0, 0, 0, 1), 'rxyx': (0, 0, 1, 1), 'ryzx': (0, 1, 0, 1), - 'rxzx': (0, 1, 1, 1), 'rxzy': (1, 0, 0, 1), 'ryzy': (1, 0, 1, 1), - 'rzxy': (1, 1, 0, 1), 'ryxy': (1, 1, 1, 1), 'ryxz': (2, 0, 0, 1), - 'rzxz': (2, 0, 1, 1), 'rxyz': (2, 1, 0, 1), 'rzyz': (2, 1, 1, 1)} - -_TUPLE2AXES = dict((v, k) for k, v in _AXES2TUPLE.items()) - -# helper functions - -def vector_norm(data, axis=None, out=None): - """Return length, i.e. eucledian norm, of ndarray along axis. - - >>> v = numpy.random.random(3) - >>> n = vector_norm(v) - >>> numpy.allclose(n, numpy.linalg.norm(v)) - True - >>> v = numpy.random.rand(6, 5, 3) - >>> n = vector_norm(v, axis=-1) - >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=2))) - True - >>> n = vector_norm(v, axis=1) - >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1))) - True - >>> v = numpy.random.rand(5, 4, 3) - >>> n = numpy.empty((5, 3), dtype=numpy.float64) - >>> vector_norm(v, axis=1, out=n) - >>> numpy.allclose(n, numpy.sqrt(numpy.sum(v*v, axis=1))) - True - >>> vector_norm([]) - 0.0 - >>> vector_norm([1.0]) - 1.0 - - """ - data = numpy.array(data, dtype=numpy.float64, copy=True) - if out is None: - if data.ndim == 1: - return math.sqrt(numpy.dot(data, data)) - data *= data - out = numpy.atleast_1d(numpy.sum(data, axis=axis)) - numpy.sqrt(out, out) - return out - else: - data *= data - numpy.sum(data, axis=axis, out=out) - numpy.sqrt(out, out) - - -def unit_vector(data, axis=None, out=None): - """Return ndarray normalized by length, i.e. eucledian norm, along axis. - - >>> v0 = numpy.random.random(3) - >>> v1 = unit_vector(v0) - >>> numpy.allclose(v1, v0 / numpy.linalg.norm(v0)) - True - >>> v0 = numpy.random.rand(5, 4, 3) - >>> v1 = unit_vector(v0, axis=-1) - >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=2)), 2) - >>> numpy.allclose(v1, v2) - True - >>> v1 = unit_vector(v0, axis=1) - >>> v2 = v0 / numpy.expand_dims(numpy.sqrt(numpy.sum(v0*v0, axis=1)), 1) - >>> numpy.allclose(v1, v2) - True - >>> v1 = numpy.empty((5, 4, 3), dtype=numpy.float64) - >>> unit_vector(v0, axis=1, out=v1) - >>> numpy.allclose(v1, v2) - True - >>> list(unit_vector([])) - [] - >>> list(unit_vector([1.0])) - [1.0] - - """ - if out is None: - data = numpy.array(data, dtype=numpy.float64, copy=True) - if data.ndim == 1: - data /= math.sqrt(numpy.dot(data, data)) - return data - else: - if out is not data: - out[:] = numpy.array(data, copy=False) - data = out - length = numpy.atleast_1d(numpy.sum(data*data, axis)) - numpy.sqrt(length, length) - if axis is not None: - length = numpy.expand_dims(length, axis) - data /= length - if out is None: - return data - - -def random_vector(size): - """Return array of random doubles in the half-open interval [0.0, 1.0). - - >>> v = random_vector(10000) - >>> numpy.all(v >= 0.0) and numpy.all(v < 1.0) - True - >>> v0 = random_vector(10) - >>> v1 = random_vector(10) - >>> numpy.any(v0 == v1) - False - - """ - return numpy.random.random(size) - - -def inverse_matrix(matrix): - """Return inverse of square transformation matrix. - - >>> M0 = random_rotation_matrix() - >>> M1 = inverse_matrix(M0.T) - >>> numpy.allclose(M1, numpy.linalg.inv(M0.T)) - True - >>> for size in range(1, 7): - ... M0 = numpy.random.rand(size, size) - ... M1 = inverse_matrix(M0) - ... if not numpy.allclose(M1, numpy.linalg.inv(M0)): print size - - """ - return numpy.linalg.inv(matrix) - - -def concatenate_matrices(*matrices): - """Return concatenation of series of transformation matrices. - - >>> M = numpy.random.rand(16).reshape((4, 4)) - 0.5 - >>> numpy.allclose(M, concatenate_matrices(M)) - True - >>> numpy.allclose(numpy.dot(M, M.T), concatenate_matrices(M, M.T)) - True - - """ - M = numpy.identity(4) - for i in matrices: - M = numpy.dot(M, i) - return M - - -def is_same_transform(matrix0, matrix1): - """Return True if two matrices perform same transformation. - - >>> is_same_transform(numpy.identity(4), numpy.identity(4)) - True - >>> is_same_transform(numpy.identity(4), random_rotation_matrix()) - False - - """ - matrix0 = numpy.array(matrix0, dtype=numpy.float64, copy=True) - matrix0 /= matrix0[3, 3] - matrix1 = numpy.array(matrix1, dtype=numpy.float64, copy=True) - matrix1 /= matrix1[3, 3] - return numpy.allclose(matrix0, matrix1) - - -def _import_module(module_name, warn=True, prefix='_py_', ignore='_'): - """Try import all public attributes from module into global namespace. - - Existing attributes with name clashes are renamed with prefix. - Attributes starting with underscore are ignored by default. - - Return True on successful import. - - """ - try: - module = __import__(module_name) - except ImportError: - if warn: - warnings.warn("Failed to import module " + module_name) - else: - for attr in dir(module): - if ignore and attr.startswith(ignore): - continue - if prefix: - if attr in globals(): - globals()[prefix + attr] = globals()[attr] - elif warn: - warnings.warn("No Python implementation of " + attr) - globals()[attr] = getattr(module, attr) - return True |