A python binding using pybind11 for Sophus, which is a C++ Lie library.(SO3 && SE3), used for 2d and 3d geometric problems (i.e. for Computer Vision or Robotics applications)
SophusPy is perfectly compatible with Numpy.
pip install sophuspyimport numpy as np
import sophuspy as sp
# 1. constructor of SO2
sp.SO2() # default
sp.SO2([[1, 0],
[0, 1]]) # list
sp.SO2(np.eye(2)) # numpy
'''
SO2([[1, 0],
[0, 1]])
'''
# 2. constructor of SO3
sp.SO3() # default
sp.SO3([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]]) # list
sp.SO3(np.eye(3)) # numpy
'''
SO3([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
'''
# 3. constructor of SE2
sp.SE2() # default
sp.SE2([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]]) # list
sp.SE2(np.eye(3)) # numpy
'''
SE2([[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
'''
# 4. constructor of SE3
sp.SE3() # default
sp.SE3([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]]) # list
sp.SE3(np.eye(4)) # numpy
'''
SE3([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]])
'''
# 5. R, t constructor of SE2
sp.SE2(np.eye(2), np.ones(2)) # R, t
'''
SE2([[1, 0, 1],
[0, 1, 1],
[0, 0, 1]])
'''
# 6. R, t constructor of SE3
sp.SE3(np.eye(3), np.ones(3)) # R, t
'''
SE3([[1, 0, 0, 1],
[0, 1, 0, 1],
[0, 0, 1, 1],
[0, 0, 0, 1]])
'''R = sp.SO3()
R1 = sp.SO3([[0, 1, 0],
[0, 0, 1],
[1, 0, 0]])
# 1. SO3 * SO3
R * R1
'''
SO3([[0, 1, 0],
[0, 0, 1],
[1, 0, 0]])
'''
# 2.
R1 *= R
T = sp.SE3()
T1 = sp.SE3(R1.matrix(), np.ones(3))
# 3. SE3 * SE3
T * T1
'''
SE3([[0, 1, 0, 1],
[0, 0, 1, 1],
[1, 0, 0, 1],
[0, 0, 0, 1]])
'''
# 4.
T1 *= TR = sp.SO2([[0, -1],
[1, 0]])
T = sp.SE2(R.matrix(), np.ones(2))
pt = np.array([1, 2])
pts = np.array([[1, 2],
[3, 4]])
# 1. single point
R * pt # array([-2., 1.])
# 2. N points
R * pts # array([[-2., 1.],
# [-4., 3.]])
# 3. single point
T * pt # array([-1., 2.])
# 4. N points
T * pts # array([[-1., 2.],
# [-3., 4.]])R = sp.SO3([[0, 1, 0],
[0, 0, 1],
[1, 0, 0]])
T = sp.SE3(R.matrix(), np.ones(3))
pt = np.array([1, 2, 3])
pts = np.array([[1, 2, 3],
[4, 5, 6]])
# 1. single point
R * pt # array([2., 3., 1.])
# 2. N points
R * pts # array([[2., 3., 1.],
# [5., 6., 4.]])
# 3. single point
T * pt # array([3., 4., 2.])
# 4. N points
T * pts # array([[3., 4., 2.],
# [6., 7., 5.]])R = sp.SO3([[0, 1, 0],
[0, 0, 1],
[1, 0, 0]])
T = sp.SE3(R.matrix(), np.ones(3))
# 1.
R.matrix()
'''
array([[0., 1., 0.],
[0., 0., 1.],
[1., 0., 0.]])
'''
# 2.
R.log() # array([-1.20919958, -1.20919958, -1.20919958])
# 3.
R.inverse()
'''
SO3([[0, 0, 1],
[1, 0, 0],
[0, 1, 0]])
'''
# 4.
R.copy()
# 5.
T.matrix()
'''
array([[0., 1., 0., 1.],
[0., 0., 1., 1.],
[1., 0., 0., 1.],
[0., 0., 0., 1.]])
'''
# 6.
T.matrix3x4()
'''
array([[0., 1., 0., 1.],
[0., 0., 1., 1.],
[1., 0., 0., 1.]])
'''
T_SE2.matrix2x3() # For SE2
# 7.
T.so3()
'''
SO3([[0, 1, 0],
[0, 0, 1],
[1, 0, 0]])
'''
# 8.
T.log() # array([1., 1., 1., -1.20919958, -1.20919958, -1.20919958])
# 9.
T.inverse()
'''
SE3([[ 0, 0, 1, -1],
[ 1, 0, 0, -1],
[ 0, 1, 0, -1],
[ 0, 0, 0, 1]])
'''
# 10.
T.copy()
# 11.
T.translation() # array([1., 1., 1.])
# 12.
T.rotationMatrix()
'''
array([[0., 1., 0.],
[0., 0., 1.],
[1., 0., 0.]])
'''
# 13.
T.setRotationMatrix(np.eye(3)) # set SO3 matrix
# 14.
T.setTranslation(np.zeros(3)) # set translationsp.SO2.hat(1)
'''
array([[ 0., -1.],
[ 1., 0.]])
'''
sp.SO3.hat(np.ones(3))
'''
array([[ 0., -1., 1.],
[ 1., 0., -1.],
[-1., 1., 0.]])
'''
sp.SO2.exp(1)
'''
SO2([[ 0.54030230586814, -0.841470984807897],
[ 0.841470984807897, 0.54030230586814]])
'''
sp.SO3.exp(np.ones(3))
'''
array([[ 0.22629564, -0.18300792, 0.95671228],
[ 0.95671228, 0.22629564, -0.18300792],
[-0.18300792, 0.95671228, 0.22629564]])
'''
sp.SE2.hat(np.ones(3))
'''
array([[ 0., -1., 1.],
[ 1., 0., 1.],
[ 0., 0., 0.]])
'''
sp.SE3.hat(np.ones(6))
'''
array([[ 0., -1., 1., 1.],
[ 1., 0., -1., 1.],
[-1., 1., 0., 1.],
[ 0., 0., 0., 0.]])
'''
sp.SE2.exp(np.ones(3))
'''
SE2([[ 0.54030230586814, -0.841470984807897, 0.381773290676036],
[ 0.841470984807897, 0.54030230586814, 1.30116867893976],
[ 0, 0, 1]])
'''
sp.SE3.exp(np.ones(6))
'''
array([[ 0.22629564, -0.18300792, 0.95671228, 1. ],
[ 0.95671228, 0.22629564, -0.18300792, 1. ],
[-0.18300792, 0.95671228, 0.22629564, 1. ],
[ 0. , 0. , 0. , 1. ]])
'''# 1. copy SO3
sp.copyto(R, R1) # copytoSO3(SO3d &dst, const SO3d &src)
# 2. copy SE3
sp.copyto(T, T1) # copytoSE3(SE3d &dst, const SE3d &src)
# 3.if R is not a strict rotation matrix, normalize it. Uses Eigen3
# Eigen::Quaterniond q(R);
# q.normalized().toRotationMatrix();
R_matrix = np.array([[1. , 0.001, 0. ],
[0. , 1. , 0. ],
[0. , 0. , 1. ]])
sp.to_orthogonal(R_matrix)
sp.to_orthogonal_3d(R_matrix) # the same as to_orthogonal
'''
array([[ 9.99999875e-01, 4.99999969e-04, 0.00000000e+00],
[-4.99999969e-04, 9.99999875e-01, -0.00000000e+00],
[-0.00000000e+00, 0.00000000e+00, 1.00000000e+00]])
'''
# if R(2D) is not a strict rotation matrix, normalize it. Uses Eigen3
# Eigen::Rotation2Dd rotation;
# rotation.fromRotationMatrix(R);
# rotation.toRotationMatrix();
sp.to_orthogonal_2d(matrix2x2) # 2D verison to_orthogonal
# 4. invert N poses in a row
pose = T.matrix3x4().ravel() # array([1., 0., 0., 0., 0., 1., 0., 0., 0., 0., 1., 0.])
sp.invert_poses(pose) # array([1., 0., 0., 0., 0., 1., 0., 0., 0., 0., 1., 0.]) identity matrix returns the same
poses = np.array([[1., 0., 0., 0., 0., 1., 0., 0., 0., 0., 1., 0.],
[0., 1., 0., 1., 0., 0., 1., 1., 1., 0., 0., 1.]])
sp.invert_poses(poses)
'''
array([[ 1., 0., 0., -0., 0., 1., 0., -0., 0., 0., 1., -0.],
[ 0., 0., 1., -1., 1., 0., 0., -1., 0., 1., 0., -1.]])
'''
# 6. transform N points by M poses to form N * M points
points = np.array([[1., 2., 3.],
[4., 5., 6.],
[7., 8., 9.]])
sp.transform_points_by_poses(poses, points)
'''
array([[ 1., 2., 3.],
[ 4., 5., 6.],
[ 7., 8., 9.],
[ 3., 4., 2.],
[ 6., 7., 5.],
[ 9., 10., 8.]])
'''