API Documentation#

`jaxtransform3d.rotations`#

Rotations in 3D#

The group of all rotations in the 3D Cartesian space is called \(SO(3)\) (SO: special orthogonal group). It is typically represented by 3D rotation matrices. The minimum number of components that are required to describe any rotation from \(SO(3)\) is 3. However, there is no representation that is non-redundant, continuous, and free of singularities.

Rotation Matrices#

`norm_matrix`(R)	Orthonormalize rotation matrix.
`robust_polar_decomposition`(A[, n_iter, eps])	Orthonormalize rotation matrix with robust polar decomposition.
`matrix_inverse`(R)	Invert rotation matrix.
`compose_matrices`(R1, R2)	Compose rotation matrices.
`apply_matrix`(R, v)	Apply rotation matrix to vector.
`matrix_from_compact_axis_angle`([axis_angle])	Compute rotation matrices from compact axis-angle representations.
`compact_axis_angle_from_matrix`(R)	Compute axis-angle from rotation matrix.

Quaternions#

`norm_quaternion`(q)	Normalize quaternion to unit norm.
`quaternion_conjugate`(q)	Conjugate of quaternion.
`compose_quaternions`(q1, q2)	Compose two quaternions.
`apply_quaternion`(q, v)	Apply rotation represented by a quaternion to a vector.
`quaternion_from_compact_axis_angle`(axis_angle)	Compute quaternion from axis-angle.
`compact_axis_angle_from_quaternion`(q)	Compute axis-angle from quaternion.

Jacobians#

`left_jacobian_SO3`(axis_angle)	Left Jacobian of SO(3) at theta (angle of rotation).
`left_jacobian_SO3_series`(axis_angle)	Left Jacobian of SO(3) at theta from Taylor series with 10 terms.
`left_jacobian_SO3_inv`(axis_angle)	Inverse left Jacobian of SO(3) at theta (angle of rotation).
`left_jacobian_SO3_inv_series`(axis_angle)	Inverse left Jacobian of SO(3) at theta from Taylor series with 10 terms.

`jaxtransform3d.transformations`#

Proper Rigid Transformations in 3D#

The group of all proper rigid transformations (rototranslations) in 3D Cartesian space is \(SE(3)\) (SE: special Euclidean group). Transformations consist of a rotation and a translation. Those can be represented in different ways just like rotations can be expressed in different ways. The minimum number of components that are required to describe any transformation from \(SE(3)\) is 6.

Transformation Matrices#

`transform_inverse`(T)	Invert transformation matrix.
`apply_transform`(T, v)	Apply transformation matrix to vector.
`compose_transforms`(T1, T2)	Compose transformation matrices.
`create_transform`(R, t)	Make transformation from rotation matrix and translation.
`transform_from_exponential_coordinates`(...)	Compute transformation matrix from exponential coordinates.
`exponential_coordinates_from_transform`(T)	Compute exponential coordinates from transformation matrix.

Dual Quaternions#

`norm_dual_quaternion`(dual_quat)	Normalize unit dual quaternion.
`dual_quaternion_squared_norm`(dual_quat)	Compute squared norm of dual quaternion.
`dual_quaternion_quaternion_conjugate`(dual_quat)	Quaternion conjugate of dual quaternions.
`compose_dual_quaternions`(dual_quat1, dual_quat2)	Concatenate dual quaternions.
`apply_dual_quaternion`(dual_quat, v)	Apply transform represented by a dual quaternion to a vector.
`dual_quaternion_from_exponential_coordinates`(...)	Compute dual quaternion from exponential coordinates.
`exponential_coordinates_from_dual_quaternion`(...)	Compute dual quaternion from exponential coordinates.