Parametric Curves: A Review

As mentioned in the discussion of boundary representations, each face is surrounded by edges, which could be line segments or curve segments, and the face itself is part of a surface (i.e., a surface patch). In this unit, we shall discuss the general concept of curve segments in parametric form.

A parametric curve in space has the following form:

f: [0,1] -> ( f(u), g(u), h(u) )
where f(), g() and h() are three real-valued functions. Thus, f(u) maps a real value u in the closed interval [0,1] to a point in space. The domain of these real functions and vector-valued function f() does not have to be [0,1]. It can be any closed interval; but, for simplicity, we restrict the domain to [0,1]. Thus, for each u in [0,1], there corresponds to a point ( f(u), g(u), h(u) ) in space.

In this course, functions f(), g() and h() are always polynomials.

Note that if function h() is removed from the definition of f(), f() has two coordinate components and becomes a curve in the coordinate plane.