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Geometry of the rolling ellipsoid. (English) Zbl 1374.53033

Kybernetika 52, No. 2, 209-223 (2016); erratum ibid. 52, No. 6, 1003-1004 (2016).
Summary: We study rolling maps of the Euclidean ellipsoid, rolling upon its affine tangent space at a point. Driven by the geometry of rolling maps, we find a simple formula for the angular velocity of the rolling ellipsoid along any piecewise smooth curve in terms of the Gauss map. This result is then generalised to rolling any smooth hyper-surface. On the way, we derive a formula for the Gaussian curvature of an ellipsoid which has an elementary proof and has been previously known only for dimension two.

MSC:

53B21 Methods of local Riemannian geometry
53A05 Surfaces in Euclidean and related spaces
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
70B10 Kinematics of a rigid body
35B06 Symmetries, invariants, etc. in context of PDEs
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