Spline curves

Abstract

Throughout this book, visual curves are represented in terms of parametric spline curves, as is common in computer graphics. These are curves (x(s),y(s)) in which s is a parameter that increases as the curve is traversed, and x and y are particular functions of s, known as splines. A spline of order d is a piecewise polynomial function, consisting of concatenated polynomial segments or spans, each of some polynomial order d, joined together at breakpoints. Parametric spline curves are attractive because they are capable of representing efficiently sets of boundary curves in an image (figure 3.1). Simple shapes can be represented by a curve with just a few spans. More complex shapes could be accommodated by raising the polynomial order d but it is preferable to increase the number of spans used. Usually the polynomial order is fixed at quadratic (d = 3) or cubic (d = 4)¹. Maintaining a fixed, low polynomial degree, even in the face of geometric complexity, makes for computational stability and simplicity.