Manifold operations

wave_geometry includes operations on \(\SO3\), the Lie group of 3D rotations, and \(\SE3\), the group of 3D rigid transformations.

Supported manifold operations

Operation		Code
Composition	\(\vgrp \circ \vgrp\)	R * R
Inverse	\(\vgrp^{-1}\)	inverse(R)
Coordinate map	\(\vgrp (\vvec)\)	R * p
Exponential map	\(\exp(\valg)\)	exp(w)
Logarithmic map	\(\log(\vgrp)\)	log(R)
Manifold plus	\(\vgrp \boxplus \valg = \exp(\valg) \circ \vgrp\)	R + w
Manifold minus	\(\vgrp_1 \boxminus \vgrp_2 = \log(\vgrp_1 \circ \vgrp_2^{-1})\)	R - R

Above, \(\vgrp\) or R represents a Lie group element, \(\valg\) or w represents a Lie algebra element, \(\vvec\) and p represents a translation. The following operations are supported for Euclidean elements, where \(\vvecg\) or v represents any element of \(\R{n}\), \(\so3\) or \(\se3\), and \(a\) or a represents a scalar:vector

Supported vector operations

Operation		Code
Sum	\(\vvecg+\vvecg\)	v + v
Difference	\(\vvecg-\vvecg\)	v - v
Negation	\(-\vvecg\)	-v
Scalar multiplication	\(a\vvecg\)	a * v
Scalar division	\(\vvecg/a\)	v / a
Dot product	\(\vvecg \cdot \vvecg\)	dot(v, a)

These operations are supported for affine points, \(\vvec\):

Supported point operations

Operation		Code
Translation between points	\(\vvec - \vvec = \vvecg\)	p - p
Point translation	\(\vvec + \vvecg = \vvec\)	p + v
Point translation	\(\vvec - \vvecg = \vvec\)	p - v