The cubic de Casteljau construction and some classes of cubic curves on Riemannian manifolds

The interpolation of data by curves in non-Euclidean spaces is important for trajectory planning, quantum computing and image registration. In particular, Riemannian cubics, Jupp and Kent cubics and Riemanninan cubics in tension are frequently used for Hermite interpolation. Except in very few cases, these curves cannot be given in closed form, which limits their applicability. The classical de Casteljau construction of cubic polynomials in Euclidean space has been generalized to Riemannian manifolds, where line segments are replaced by geodesic arcs on manifold. The authors (Zhang and Noakes, 2019a) showed that generalized cubic de Casteljau curves, can approximate Riemannian cubics quite closely, with error bounded by a 4th order curvature term. Motivated by that work, the present paper aims to analyze the quality of the generalized cubic de Casteljau curves approximating Jupp and Kent cubics, and Riemannian cubics in tension. We also modify the generalized de Casteljau algorithm to construct curves that are closer to Jupp and Kent cubics, and to Riemannian cubics in tension. We illustrate our theoretical results by numerical experiments on cubic curves in the 2-dimensional unit sphere $S^2$ and also in the special orthogonal group $\mathrm{SO}\left(3\right)$.