On the Geometry of Rolling and Interpolation Curves on Sn, SOn, and Grassmann Manifolds

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