Geometric design of motions constrained by a contacting surface pair

We discuss the following problem which arises in robot motion planning, NC machining and computer animation: Given are a fixed surface Ψ and N positions Φi of a moving surface Φ such that the Φi are in point contact with Ψ. Compute a smooth and fair Euclidean gliding motion Φ(t) of the surface Φ on the surface Ψ which interpolates (or approximates) the given positions Φi at time instances ti. First we generalize interpolatory variational subdivision algorithms for curves to curves on surfaces. Second we study an unconstraint motion design algorithm which we then extend to the main contribution of this paper, an algorithm for the design of a motion constraint by a contacting surface pair. Both motion design algorithms use a feature point representation of the moving surface, subdivision algorithms for curves, instantaneous kinematics, and ideas from line geometry. Geometric methods are used for the numerical solution of the arising optimization problems.