#!/usr/bin/env python # -*- 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 (devided 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