Bézier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications

In this article we generalize the concept of Bezier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau's algorithm for constructing Bezier 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 Bezier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm 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