Generators for Pairs of Position-Orientations

In our group, we frequently work with data on position-orientation space \(\mathbb{M}_d := \mathbb{R}^d \times S^{d - 1}\). Position-orientation space \(\mathbb{M}_d\) is naturally acted on by the roto-translation group \(\operatorname{SE}(d)\): \[ g := (y, R) \in \operatorname{SE}(d), p := (x, n) \in \mathbb{M}_d \mapsto g p := (y + R x, R n). \] It can therefore be desirable to perform processing on \(\mathbb{M}_d\) that is equivariant to \(\operatorname{SE}(d)\), which essentially means that roto-translations commute with the processing.

A particularly interesting situation is when we want to process point clouds of position-orientations. Suppose we are looking for equivariant functions (e.g. neural networks) \(\Phi: \mathbb{M}_d \to \mathbb{R}\). It is not so hard to see that this permits only constant functions \(\Phi\): since \(\mathbb{M}_d\) is a homogeneous space of \(\operatorname{SE}(d)\), for any \(p_1, p_2 \in \mathbb{M}_d\), there exists a \(g \in \operatorname{SE}(d)\) such that \(g p_1 = p_2\), so necessarily \(\Phi(p_1) = \Phi(p_2)\).

Since this is rather restrictive, we now look instead for equivariant functions \(\Phi: \mathbb{M}_d \times \mathbb{M}_d \to \mathbb{R}\). Fortunately, there are actually non-trivial equivariant scalar functions on pairs of position-orientations. In Roto-Translation Invariant Metrics on Position-Orientation Space, Bellaard & Smets classify a whole family of such functions in the case \(d = 3\). These functions are the lengths of specific exponential curves connecting pairs of position-orientations: we get different functions by varying the parameters of the \(\operatorname{SE}(3)\)-invariant metric. An exponential curve \(\eta: [0, 1] \to \mathbb{M}_3\) is then \[ \eta(t) := \exp(t M) p \] for some position-orientation \(p \in \mathbb{M}_3\) and generator \(M \in \mathfrak{se}(3)\); its length can easily be computed using \[ L_\mathcal{G}(\eta) = \|M p\|_\mathcal{G}, \] where \(\mathcal{G}\) is the \(\operatorname{SE}(3)\)-invariant metric. It turns out that for any pair \(p_1, p_2 \in \mathbb{M}_3\) there exist many generators \(M(p_1, p_2)\) such that \(\eta(0) = p_1\) and \(\eta(1) = p_2\), with \(\eta(t) := \exp(t M(p_1, p_2)) p_1\).

In Roto-Translation Invariant Metrics on Position-Orientation Space, Bellaard & Smets pick the generator with the minimal angular velocity \(M_\mathrm{mav}\). This corresponds to the exponential curve which rotates the least. Another interesting choice is the pure rotation generator \(M_\mathrm{rot}\), which connects the pair by only rotating around some point. Generically, the collection of all possible connecting generators is diffeomorphic to the circle \(S^1\). It is therefore possible to parametrise all such generators using an angle \(\phi \in [-\pi, \pi)\), where by convention \(M_0 := M_\mathrm{mav}\).

In this visualisation, we show a pair of position-orientations \(p_1 := (x_1, n_1), p_2 := (x_2, n_2) \in \mathbb{M}_3\) in black, and three connecting exponential curves: \[ \begin{align*} \textcolor{#B6321C}{\eta_\mathrm{mav}(t)} & := \exp(t M_\mathrm{mav}(p_1, p_2)) p_1, \\ \textcolor{#009B55}{\eta_\mathrm{rot}(t)} & := \exp(t M_\mathrm{rot}(p_1, p_2)) p_1, \\ \textcolor{#0071BC}{\eta_\phi(t)} & := \exp(t M_\phi(p_1, p_2)) p_1. \end{align*} \]

The position-orientations \(p_1, p_2\), shown as black arrows, and the generator parameter \(\phi\) can be edited in the form below; press "Submit" to update these values in the visualisation. The camera is centred on the origin \((0, 0, 0)\), and can be rotated around the origin by dragging over the canvas. Scrolling up and down over the canvas will increase and decrease the rotation speed, respectively.

The camera has been adapted from bellaard.com to be exclusively an orbit camera.