Control point based exact description of curves and surfaces in extended Chebyshev spaces

TL;DR

This paper develops explicit formulas for transformation matrices and control point configurations to exactly represent curves and surfaces in extended Chebyshev spaces, facilitating precise modeling in geometric design.

Contribution

It introduces explicit formulas for transformation matrices and control points for exact curve and surface representation in extended Chebyshev spaces, including rational variants.

Findings

Explicit formulas for transformation matrices derived.

Control point configurations for exact curve/surface representation provided.

Methods applicable to polynomial, trigonometric, hyperbolic, and mixed spaces.

Abstract

Extended Chebyshev spaces that also comprise the constants represent large families of functions that can be used in real-life modeling or engineering applications that also involve important (e.g. transcendental) integral or rational curves and surfaces. Concerning computer aided geometric design, the unique normalized B-bases of such vector spaces ensure optimal shape preserving properties, important evaluation or subdivision algorithms and useful shape parameters. Therefore, we propose global explicit formulas for the entries of those transformation matrices that map these normalized B-bases to the traditional (or ordinary) bases of the underlying vector spaces. Then, we also describe general and ready to use control point configurations for the exact representation of those traditional integral parametric curves and (hybrid) surfaces that are specified by coordinate functions given as (products of separable) linear combinations of ordinary basis functions. The obtained results are also extended to the control point and weight based exact description of the rational counterpart of these integral parametric curves and surfaces. The universal applicability of our methods is presented through polynomial, trigonometric, hyperbolic or mixed extended Chebyshev vector spaces.