Embedding-based Interpolation on the Special Orthogonal Group

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 × 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 C1-continuity.