Ágoston Róth

Ágoston Róth

We generalize the transfinite triangular interpolant of (Nielson, 1987) in order to generate visually smooth (not necessarily polynomial) local interpolating quasi-optimal triangular spline surfaces. Given as input a triangular mesh stored in a half-edge data structure, at first we produce a local interpolating network of curves by optimizing quadratic energy functionals described along the arcs as weighted combinations of squared length variations of first and higher order derivatives, then by optimizing weighted combinations of first and higher order quadratic thin-plate-spline-like energies we locally interpolate each curvilinear face of the previous curve network with triangular patches that are usually only $C^0$ continuous along their common boundaries. In a following step, these local interpolating optimal triangular surface patches are used to construct quasi-optimal continuous vector fields of averaged unit normals along the joints, and finally we extend the $G^1$ continuous transfinite triangular interpolation scheme of (Nielson, 1987) by imposing further optimality constraints concerning the isoparametric lines of those groups of three side-vertex interpolants that have to be convexly blended in order to generate the final visually smooth local interpolating quasi-optimal triangular spline surface. While we describe the problem in a general context, we present examples in special polynomial, trigonometric, hyperbolic and algebraic-trigonometric vector spaces of functions that may be useful both in computer-aided geometric design and in computer graphics.

Ágoston Róth

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 constant-coefficient 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 so-called 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.

Ágoston Róth

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.