Control point based exact description of higher dimensional trigonometric and hyperbolic curves and multivariate surfaces

Abstract: Using the normalized B-bases of vector spaces of trigonometric and hyperbolic polynomials of finite order, we specify control point configurations for the exact description of higher dimensional (rational) curves and (hybrid) multivariate surfaces determined by coordinate functions that are exclusively given either by traditional trigonometric or hyperbolic polynomials in each of their variables. The usefulness and applicability of theoretical results and proposed algorithms are illustrated by many examples that also comprise the control point based exact description of several famous curves (like epi- and hypocycloids, foliums, torus knots, Bernoulli's lemniscate, hyperbolas), surfaces (such as pure trigonometric or hybrid surfaces of revolution like tori and hyperboloids, respectively) and 3-dimensional volumes. The core of the proposed modeling methods relies on basis transformation matrices with entries that can be efficiently obtained by order elevation. Providing subdivision formulae for curves described by convex combinations of these normalized B-basis functions and control points, we also ensure the possible incorporation of all proposed techniques into today's CAD systems.