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On the geometry of rolling and interpolation curves on \(S^n\), \(\mathrm{SO}_n\), and Grassmann manifolds. (English) Zbl 1140.58005
Summary: 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 \(\text{SO}_{n}\), 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.

MSC:
58E40 Variational aspects of group actions in infinite-dimensional spaces
53A17 Differential geometric aspects in kinematics
53B05 Linear and affine connections
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
93C15 Control/observation systems governed by ordinary differential equations
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