57d4c86b2b
Also, removed the one from modules/python/src2/cv.py and cleared its executable bit, since it's not a script.
1708 lines
56 KiB
Python
Executable File
1708 lines
56 KiB
Python
Executable File
#!/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
|