9.2. Parametric Curves   

Parametric polynomial curves define points on a 3D curve by using three polynomials in a parameter t, one for each of x, y, z. If the parameter is thought of as time, the parametric function is used to locate the particle in space at a given instant. As time passes, the particle sweeps out a path, thereby tracing a curve. A parametric function therefore defines more that just a path. There is also information about the direction and speed of the particle as it moves along the path. (Barsky and DeRose, 1989)

A parametric spline function is a piecewise function where each of the pieces is a parametric function. The pieces of a curve are known as segments, while those of a surface are called patches. An important aspect of these functions is the way the segments are joined together. The locations where the pieces of the function join are called joints in the case of curves, and borders in the case of surfaces. The equations that govern this joining are called continuity constraints.

In computer-aided geometric design, the continuity constraints are typically chosen to impart a given order of smoothness to the spline. The order of smoothness chosen will naturally depend on the application. For some applications, such as architectural drawing, it is sufficient for the curves to be continuous only in position. Other applications, such as the design of mechanical parts, require first or second-order smoothness.

In fact, there is more than one type of smoothness. For instance, if parametric splines are being used to define the path of an object in an animation system, the object must move smoothly. It is therefore not only for the path of the object to be smooth, the speed of the object as it moves along the path must also be continuous.

This type of motion can be guaranteed by requiring continuity of position and the first parametric derivative vector, also known as the velocity vector. If higher order continuity is required, we can demand continuity of second parametric derivative, or acceleration vector. Most parametric spline formulations are based on parametric continuity (Barsky and DeRose, 1989).