On the Geometry of Rolling and Interpolation Curves on S ⁿ, SO _n , and Grassmann Manifolds

Abstract

We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm easy to implement. The idea is to project the prescribed data on the manifold onto the affine tangent space at a particular point, solve the interpolation problem on this affine subspace, and then project the resulting curve back on the manifold. One of the novelties of this approach is the use of rolling mappings. The manifold is required to roll on the affine subspace like a rigid body, so that the motion is described by the action of the Euclidean group on the embedding space. The interpolation problem requires a combination of a pullback/push forward with rolling and unrolling. The rolling procedure by itself highlights interesting properties and gives rise to a new, but simple, concept of geometric polynomial curves on manifolds. This paper is an extension of our previous work, where mainly the 2-sphere case was studied in detail. The present paper includes results for the n-sphere, orthogonal group SO _n , and real Grassmann manifolds. In particular, we present the kinematic equations for rolling these manifolds along curves without slip or twist, and derive from them formulas for the parallel transport of vectors along curves on the manifold.