Equilibrium and diffusion equations using cylindroidal coordinates

Presented is a system of curvilinear coordinates based on a cylindroid. Cartesian, cylindrical and spherical coordinates are special scenarios that emerge from a system of cylindroidal coordinates. A system of cylindroidal coordinates was originally proposed to parameterize toothed bodies or generalized hyperboloidal gear elements, and consequently certain fundamental relations for conjugate hyperboloidal pitch surfaces in mesh are presented. Two applications of cylindroidal coordinates are presented. The first application addresses the equilibrium equation and the second application addresses the diffusion equation for a differential cylindroidal element. It is demonstrated that the developed diffusion equation degenerates into established diffusion relations for cylindrical, spherical and Cartesian coordinates.