## 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).