Control point based curve and surface modeling in extended Chebyshev spaces

Abstract

By means of a basic software application, we would like to demonstrate the practical use of B-curves and B-surfaces defined over extended Chebyshev spaces (EC spaces) in control point based graphical modeling.

Using existing theoretical results we present numerical methods for computing the (zeroth and higher order) derivatives of the blending functions needed for generating EC B-curves and surfaces as well as algorithms for the general order elevation and subdivision of these. A basis transformation between the ordinary basis and the unique normalized B-basis (NB-basis) of an EC space is also presented, making control point based exact description possible for curves and surfaces given in a parametric form.

The application is written in C++ using the OpenGL and OpenMP libraries as well as the Qt Widgets framework. It provides the possibility of handling any EC space that can be identified by the solution space of a constant-coefficient homogeneous linear differential equation defined by the user. The multi- threaded implementation of the algorithms mentioned is numerically stable and efficient in practice for spaces with up to a sufficiently large dimension number.

I would like to thank my advisor, Ágoston Róth (https://sites.google.com/site/agostonroth), who was of great help to me both in understanding and applying the theoretical results and in implementing the necessary numerical methods and OpenGL related functionalities as well.

Downloads:

This is my BSc diploma thesis. Published here under CC BY 4.0.