Mathematical Structures for Computer Graphics

Chapter 7Curves and Surfaces

A straight line segment is a key primitive in computer graphics and, consequently, any flat triangular (or more generally, polygonal) surface is easy to draw on the screen. Yet, curves are indispensable when designing objects like cars, door knobs, archways, and any number of animated characters. Although we end up approximating these forms with many triangles with their straight edges, the problem is still how to find vertices that ultimately give a global look of curvature. If we add the animation stage where we let a car bounce along the dirt roadway or move the camera as though we are flying through the city, then the paths describing these motions rely on various curves.

There are several ways to describe curves, but in order to display them on a computer screen we eventually need coordinates for various points on the curve. Connecting the points with line segments then completes the approximation of the curve (Figure 7.1). An implicit algebraic description of a circle, such as , is familiar and compact, but to find point coordinates we have to solve for one coordinate in terms of another. An alternative description, and , relies on computing trigonometric ...