Embedding-Based Interpolation on the Special Orthogonal Group

TL;DR

This paper introduces a new interpolation method for functions on the special orthogonal group $SO(n)$ using embedding and projection techniques, enabling efficient and accurate computations of $SO(n)$-valued functions and their derivatives.

Contribution

The paper proposes a novel embedding-based interpolation scheme for $SO(n)$ that preserves accuracy and regularity, with explicit evaluation formulas and connections to geodesic finite elements.

Findings

Interpolation schemes are efficient and accurate.

Methods facilitate computation of derivatives and minimum acceleration curves.

Applicable to $SO(n)$ and $SO(3)$ with quaternion representation.

Abstract

We study schemes for interpolating functions that take values in the special orthogonal group $SO(n)$. Our focus is on interpolation schemes obtained by embedding $SO(n)$ in a linear space, interpolating in the linear space, and mapping the result onto $SO(n)$ via the closest point projection. The resulting interpolants inherit both the order of accuracy and the regularity of the underlying interpolants on the linear space. The values and derivatives of the interpolants admit efficient evaluation via either explicit formulas or iterative algorithms, which we detail for two choices of embeddings: the embedding of $SO(n)$ in the space of $n \times n$ matrices and, when $n=3$, the identification of $SO(3)$ with the set of unit quaternions. Along the way, we point out a connection between these interpolation schemes and geodesic finite elements. We illustrate the utility of these interpolation schemes by numerically computing minimum acceleration curves on $SO(n)$, a task which is handled naturally with $SO(n)$-valued finite elements having $C^1$-continuity.