BEZIER CURVES ON RIEMANNIAN-MANIFOLDS AND LIE-GROUPS WITH KINEMATICS APPLICATIONS

In this article we generalize the concept of Bezier curves to curved s paces, and illustrate this generalization with an application in kinem atics. We show how De Casteljau's algorithm for constructing Bezier cu rves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Beca use of their group structure Lie groups admit an elegant, efficient re cursive algorithm for constructing Bezier curves. Spatial displacement s of a rigid body also form a Lie group, and can therefore be interpol ated (in the Bezier sense) using this recursive algorithm. We apply th is algorithm to the kinematic problem of trajectory generation or moti on interpolation for a moving rigid body. The orientation trajectory o f motions generated in this way have the important property of being i nvariant with respect to choices of inertial and bodyfixed reference f rames.