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An algorithm based on rolling to generate smooth interpolating curves on ellipsoids. (English) Zbl 1335.65023
The paper brings an algorithm generating \(C^2\) smooth interpolating curves on \(n\)-dimensional ellipsoids given in explicit form. The presented algorithm is a generalisation of the algorithm for generating interpolating curves on manifolds embedded in Euclidean space. The method described is based on a rolling motions of the ellipsoids without slipping or twisting. The ellipsoid is rolling over its affine tangent space at a point along geodesic curves. To obtain the geodesics in explicit form and ensure easy solution of the kinematics, the ellipsoids are embedded in a \((n+1)\)-dimensional space equipped with an appropriate non-Euclidean metric. Moreover, it is possible to obtain a closed form of interpolating curve on ellipsoids. Simulations and considerations about extension of the suggested algorithm to generate \(C^k\) smooth interpolating curve on any smooth manifold are given, too.

MSC:
65D17 Computer-aided design (modeling of curves and surfaces)
65D05 Numerical interpolation
65D07 Numerical computation using splines
65D10 Numerical smoothing, curve fitting
53B21 Methods of local Riemannian geometry
53C22 Geodesics in global differential geometry
70B10 Kinematics of a rigid body
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