TABLE OF CONTENTS We all have a strong intuitive sense of what a curve is. Although we never see a curve floating around free of any object, we can readily identify the curved edges and silhouettes of objects and easily imagine the curve that describes the path of a moving object. This chapter explores the mathematical definition of a curve in a form that is very useful to geometric modeling and other computer graphics applications: that definition consists of a set of parametric equations. The mathematics of parametric equations is the basis for B zier, NURBS, and Hermite curves. The curves discussed in this chapter may be placed in the Hermite family of curves. B zier curves are the subject of the next chapter, and NURBS are best left for more advanced texts on geometric modeling. Both plane curves and space curves are introduced here, followed by discussions of the tangent vector, blending functions, conic curves, reparameterization, and continuity and composite curves.
A parametric curve is one whose defining equations are given in terms of a single, common, independent variable called the parametric variable. We have already encountered parametric variables in earlier discussions of vectors, lines, and planes.
Imagine a curve in three-dimensional space. Each point on the curve has a unique set of coordinates: a specific x value, y value, and z value. Each coordinate is controlled by a separate parametric equation, whose general form looks like
where x( u) stands...
