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.

Rolling Riemannian Manifolds to Solve the Multi-class Classification Problem

“…The novelty in this alternate school of thought is that the manifold will be firstly rolled (without slip and twist) as a rigid body, then the given data is unwrapped onto the affine tangent space, where the classification is performed. For the sake of brevity the proof of concept will be done by testing with a multi-class LogitBoost algorithm [27,7] on the Grassmann manifold [6,25,29,28,12,16,24]. We remark that our paradigm is also valid with others Riemannian manifolds.…”

“…In the past few years there has been a growing interest in describing mathematically rolling motions, without slip and twist, of smooth manifolds (due to its analytic and geometric richness) [21,15,22,16]. The study of these kinematic problems proved to be relevant, in part because the knowledge on how to realize such virtual movements allows to solve complicated problems on certain manifolds, by reducing them to similar problems on much simpler manifolds.…”

“…The idea is to project all the data from the manifold onto an affine tangent space at a particular point. To mitigate the distortion induced by local diffeomorphisms, we introduce for the first time in the computer vision community a well-founded mathematical concept, so-called Rolling map [21,16]. The novelty in this alternate school of thought is that the manifold will be firstly rolled (without slipping or twisting) as a rigid body, then the given data is unwrapped onto the affine tangent space, where the classification is performed.…”

Cylindrical Surface Reconstruction by Fitting Paths on Shape Space

“…Path fitting on manifolds has been previously addressed in the literature with different approaches and for various purposes. Generic path fitting methods on manifolds include splines on manifolds [10], rolling procedures [7], subdivision schemes [4], gradient descent [13], and geodesic finite elements [14]. Interpolation of rotations (where the manifold M is the special orthogonal group SO(3)) is useful in robotics for motion planning of rigid bodies and in computer graphics for the animation of 3D objects [11].…”

Interpolation in special orthogonal groups

“…These algorithms, however, often do not generalize to higher-dimensional manifolds. Recently the authors of [7] proposed to combine the pull back/push forward technique with rolling a manifold (e.g. sphere, SO(n) or a Graßmann manifold) on its affine tangent space like a rigid body.…”