TABLE OF CONTENTS The B zier curve is an important part of almost every computer-graphics illustration program and computer-aided design system in use today. It is used in many ways, from designing the curves and surfaces of automobiles to defining the shape of letters in type fonts. And because it is numerically the most stable of all the polynomial-based curves used in these applications, the B zier curve is the ideal standard for representing the more complex piecewise polynomial curves.
In the early 1960s, Peter B zier (pronounced bay-zee-aye) began looking for a better way to define curves and surfaces, one that would be useful to a design engineer. He was familiar with the work of Ferguson and Coons and their parametric cubic curves and bicubic surfaces. However, these did not offer an intuitive way to alter and control shape. The results of B zier's research led to the curves and surfaces that bear his name and became part of the UNISURF system. The French automobile manufacturer, Renault, used UNISURF to design the sculptured surfaces of many of its products.
This chapter begins by describing a surprisingly simple geometric construction of a B zier curve, followed by a derivation of its algebraic definition, basis functions, control points, degree elevation, and truncation. It concludes by showing how to join two curves end-to-end to form a single composite curve.
We can draw a B zier curve using a simple recursive geometric construction. Let's begin by constructing a second-degree curve (Figure 15.1). We select three points
