Non-Polynomial Functions are found Comparable to Bezier Functions

S. Tannouri, W. Lupton, and A. Tannouri (USA)

Keywords

First and second order continuities, Meshes,linear interpolation used for entities of type vectors.

Abstract

In computer graphics, modeling objects with non uniform surfaces is based on geometrical construction techniques and on some mathematical functions that satisfy a number of constraints and properties. During the second half of the twentieth century, many techniques were developed and found useful to construct curved surfaces. Spline curves, NURBS(1), Hermits, and other techniques are all very well known methods to generate curves and surfaces. The most popular and efficient methods are based on Bezier and other researchers in the mathematical graphics modeling field. Bezier Curves are currently used by most modeling companies around the world; they are taught in details in all computer graphic courses. However some students find the concept of these curves not friendly enough to understand and use in their projects. In fact the derivation of Bezier functions from the geometry of Casteljau's tangents is not a simple and straightforward procedure. Our goal in this paper is to find a simpler way capable of doing what Bezier functions are capable of doing. In fact, we found non-polynomial functions but very comparable to Bezier's ones. The students understand these new functions rapidly, and can apply them successfully to model surfaces in their projects. These new curves are found to be easier than all the traditionally known ones. Despite their simplicity and efficiency, the new curves are just comparable not faster than Bezier curves, but are better for modeling and manipulating surfaces interactively, easier to understand and use in applications. This paper shows a new way of thinking that led to find new functions that parallel Bezier curves. In this paper, we compare the new curves to the classical curves of Bezier. We will present along with the curve functions their use to construct surface meshes. It is easier to use the demonstration program (not provided with this paper) than reading this paper. (1) Non-Uniform Rational B-spline Surfaces