A geometric theory of contact for computer-aided geometric design and kinematic design

This dissertation develops a theory of contact for piecewise parametric curves in Computer Aided Geometric Design and point trajectories in Kinematic Design based on the differential geometry of evolutes, polar curves, and binormal indicatrices. This theory is completely geometric, independent of parameterization and generalizes to any order. We first introduce the concepts of the nth polar curve, the nth principal evolute, and the nth binormal indicatrix and study their differential geometry. Two sets of dimensionless characteristic numbers describing the local geometry of a curve up to the nth order are defined. These characteristic numbers can be used to describe conditions for higher order contacts in an algebraic fashion for both piecewise parameteric curves in CAGD and point trajectories in Kinematic Design. The same characteristic numbers are caused to interpret contact conditions of piecewise parametric curves up to nth order in terms of the geometry of higher evolutes and binormal indicatrices. We also use the same characteristic numbers to express the contact conditions of point trajectories in kinematic design up to nth order in terms of the instantaneous invariants of the kinematic motion. The explicit coordinate independent expressions for the instantaneous invariants of spatial kinematics are provided. As illustration of the theory of contact and the determination of instantaneous invariants, a few examples are studied.