On Computing Contact Configurations of a Curved Chain

Abstract

Given a simple generalized polygon $\text{A}$ of line segments and arcs that is free to move and rotate and an oriented monotone chain $\text{B}$ composed of smooth parametric curved edges, the positions and orientations for $\text{A}$ to gouge-freely contact $\text{B}$ (i.e., the contact configurations) is a C⁰ continuous surface in a three dimensional space R³. Past results either limit $\text{B}$ to be polygonal or depend on the very complicated cylindrical algebraic decomposition algorithm, which is difficult to implement in practice and does not apply to parametric curves. We address this problem by conducting a rigorous study of the geometric and topological structures of the contact configurations surface and providing the exact mathematical descriptions of the faces, edges, and vertices on this surface. A practical intersection algorithm is proposed for computing the critical curves on the contact configurations surface. In addition, an application of the contact configurations in mill-turn machining is presented.