ABSTRACT
This chapter presents the mathematical foundations of parametric curves, which in both representation and in implementation provides a level of control and expression that approaches Klee's point in motion. The functional approach to constructing curves may initially strike a designer as an unintuitive one, as it is quite unlike that of constructing curves via control points. The variations and combinations within design space of curves is potentially endless, and the careful shaping of variables to coax a curve into a productive and useful tool for design. It is one thing to create a curve by transcribing a found parameterization from mathematical source material, and quite another to shape a curve to meet the often non-mathematical demands of a design application. The tangent vector at a curve point measures the rate of change of curve. Unlike a tangent vector at a curve point whose direction is uniquely defined by the curve parameterization, there are many possible normal vectors to a curve.
