Lishen1 / manif-rs

A small library for Lie theory

manif

A small library for Lie theory

Unofficial partial port of manif

Warning WIP

TODO

• impl SO2
• impl SE2
• impl SE3
• move common tests to macroses
• move some impl into trait defaults
• add docs

Available Operations

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

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

Jacobians

All operations come with their respective analytical Jacobian matrices. Throughout manif, Jacobians are differentiated with respect to a local perturbation on the tangent space. These Jacobians map tangent spaces, as described in this paper.

Currently, manif implements the right Jacobian, whose definition reads:

$\frac{\delta f(\mathbf{\mathcal{X}})}{\delta\mathbf{\mathcal{X}}}\triangleq\displaystyle{\lim_{\mathbf\varphi\to0}}\frac{f(\mathcal{X}\oplus\mathbf\varphi)\ominus f(\mathcal{X})}{\mathbf\varphi}\triangleq \displaystyle{\lim_{\mathbf\varphi\to0}}\frac{log(f(\mathcal{X})^{-1}f(\mathcal{X}\exp(\mathbf\varphi^\wedge)))^{\vee}}{\mathbf\varphi}$

The Jacobians of any of the aforementionned operations can then be evaluated:

License

GNU GPLv3