An OpenGL and C++ based function library for curve and surface modeling in a large class of extended Chebyshev spaces

Abstract

We propose a platform-independent multi-threaded function library that provides data structures to generate, differentiate and render both the ordinary basis and the normalized B-basis of a user-specified extended Chebyshev (EC) space that comprises the constants and can be identified with the solution space of a constantcoefficient homogeneous linear differential equation defined on a sufficiently small interval. Using the obtained normalized B-bases, our library can also generate, (partially) differentiate, modify and visualize a large family of socalled B-curves and tensor product B-surfaces. Moreover, the library also implements methods that can be used to perform dimension elevation, to subdivide B-curves and B-surfaces by means of de Casteljau-like B-algorithms, and to generate basis transformations for the B-representation of arbitrary integral curves and surfaces that are described in traditional parametric form by means of the ordinary bases of the underlying EC spaces. Independently of the algebraic, exponential, trigonometric or mixed type of the applied EC space, the proposed library is numerically stable and efficient up to a reasonable dimension number and may be useful for academics and engineers in the fields of Approximation Theory, Computer Aided Geometric Design, Computer Graphics, Isogeometric and Numerical Analysis. Keywords extended Chebyshev spaces • constant-coefficient homogeneous linear differential equations • normalized B-basis • B-curve/surface modeling • order (dimension) elevation • subdivision (B-algorithm) • basis transformation • control point based exact description • OpenGL • OpenMP Mathematics Subject Classification (2000) 65D17 • 68U07