Spatial point contact kinematics and parallel transport

Abstract

A point contact kinematic model is developed, using a non-holonomic parameterization first introduced in 1885 by Neumann and Richter. The model is built with two auxiliary pairs of imaginary bodies. Each body carries a coordinate reference frame. By this, one single point pair becomes a serial composite of three pairs, two Levi-Civita and one revolute pair. Convenient, insightful and computationally attractive algebraic expressions are obtained. The approach is general and applicable to any pair of regular, parametric surfaces. Algebra is simplified by the discovery of an intimate relationship between point-contact kinematics and parallel transport of vector fields in the sense of Levi-Civita. A new special motion, Levi-Civita motion, is defined and discussed. The following first- and second-order kinematic properties of Levi-Civita motion are presented: motion screw, motion pitch, Hamilton cylindroid, Sturm theorem, parallelism, Schieldrop-Johnsen vector of non-holonomic deviation, analogy to a rigid body with a fixed point and a kinematic proof of the Gauss-Bonnet theorem of differential geometry.