Splines in interactive computer graphics

Abstract

Computer graphics, particularly interactive computer graphics, is not, as the name might imply, concerned with drawing graphs, but rather with the broadest issues of manipulating, transforming, and displaying information in visual format. It is interactive in so far as operations can be carried out in real time — which requires algorithms of high computational efficiency and low complexity.

Splines are a valuable tool in graphics, but they are often applied in a way not used by the mathematician. This difference raises computational issues which the numerical analyst might otherwise never see. This talk will provide a brief introduction to such issues and follow with a study of two current developments.

We begin with a review of the graphics environment, mentioning the modelling and display process and pointing out some of the costly issues. The novel use of splines in interactive graphics comes through the construction of surfaces as weighted averages of selected points, called “control vertices” in which B-splines are taken as the weighting functions. Some examples will illustrate the characteristics of this use of B-splines.

With this background we consider two recent developments. The first is the control-vertex recurrence of Riesenfeld, Cohen, and Lyche; the second is Barsky’s work on geometric vs. mathematical continuity, and his introduction of Beta-splines. We will close with some results on current research concerned with a synthesis of these two developments.