Operations on Lie groups¶

manif supports the following operations with Lie groups,

Operation		Code
	Base Operation
Inverse	\(\bf\mathcal{X}^{-1}\)	`X.inverse()`
Composition	\(\bf\mathcal{X}\circ\bf\mathcal{Y}\)	`X * Y` `X.compose(Y)`
Hat	\(\boldsymbol\varphi^\wedge\)	`w.hat()`
Act on vector	\(\bf\mathcal{X}\circ{\bf v}\)	`X.act(v)`
Retract to group element	\(\exp(\boldsymbol\varphi^\wedge)\)	`w.exp()`
Lift to tangent space	\(\log(\bf\mathcal{X})^\vee\)	`X.log()`
Manifold Adjoint	\(\mathrm{Adj}(\bf\mathcal{X})\)	`X.adj()`
Tangent adjoint	\(\mathrm{adj}(\boldsymbol\varphi^\wedge)\)	`w.smallAdj()`
	Composed Operation
Manifold right plus	\({\bf\mathcal{X}}\circ\exp(\boldsymbol\varphi^\wedge)\)	`X + w` `X.plus(w)` `X.rplus(w)`
Manifold left plus	\(\exp(\boldsymbol\varphi^\wedge)\circ\bf\mathcal{X}\)	`w + X` `w.plus(X)` `w.lplus(X)`
Manifold right minus	\(\log(\bf\mathcal{Y}^{-1}\circ\bf\mathcal{X})^\vee\)	`X - Y` `X.minus(Y)` `X.rminus(Y)`
Manifold left minus	\(\log(\bf\mathcal{X}\circ\bf\mathcal{Y}^{-1})^\vee\)	`X.lminus(Y)`
Between	\({\bf\mathcal{X}^{-1}}\circ{\bf\mathcal{Y}}\)	`X.between(Y)`
Inner Product	\(\langle\boldsymbol\varphi,\boldsymbol\tau\rangle\)	`w.inner(t)`
Norm	\(\left\lVert\boldsymbol\varphi\right\rVert\)	`w.weightedNorm()` `w.squaredWeightedNorm()`

Above, \({\bf\mathcal{X}}\) & \({\bf\mathcal{Y}}\) (X & Y) represent group elements, \({\boldsymbol\varphi^\wedge}\) & \({\boldsymbol\tau^\wedge}\) represent elements in the Lie algebra of the Lie group, \({\boldsymbol\varphi}\) & \({\boldsymbol\tau}\) (w & t) represent the same elements of the tangent space but expressed in Cartesian coordinates in \(\mathbb{R}^n\), and \(\mathbf{v}\) (v) represents any element of \(\mathbb{R}^n\).