Bézier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications

[+] Author and Article Information
F. C. Park

Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA

B. Ravani

Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA

J. Mech. Des 117(1), 36-40 (Mar 01, 1995) (5 pages) doi:10.1115/1.2826114 History: Received January 01, 1994; Revised September 01, 1994; Online January 25, 2008


In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bézier sense) using this recursive algorithm. We apply this alogorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body. The orientation trajectory of motions generated in this way have the important property of being invariant with respect to choices of inertial and body-fixed reference frames.

Copyright © 1995 by The American Society of Mechanical Engineers
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