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 talk about the general concept of curve segments in parametric forms.

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 mapping a real value u in the closed interval [0,1] to real values. 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 purpose, we restrict the domain to [0,1]. Thus, 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 on the coordinate plane.

Examples

We have seen that a straight line is defined as B+td, where B is a base point and d is a direction vector. Thus, if f() is defined as
f(u) = b₁ + ud₁
g(u) = b₂ + ud₂
h(u) = b₃ + ud₃

where B = < b₁, b₂, b₃ > and d = < d₁, d₂, d₃ >. Then, f() is a parametric curve mapping [0,1] to the line segment between B and B+d inclusive.
A circle has the following non-polynomial form:
x(u) = rcos(2*PI*u) + p
y(u) = rsin(2*PI*u) + q

It has center (p, q) and radius r. Since the parameter is in the range of 0 and 1, the value of u is extended to 0 and 2*PI by multiplying 2*PI.
Let us eliminate u. First, change the above equations to the following. For convenience, we drop (u) from x(u) and y(u).
x - p = rcos(2*PI*u)
y - q = rsin(2*PI*u)

Then, squaring both equations and adding them together yields
(x - p)² + (y - q)² = r²
Thus, it shows that the given parametric form is a circle.
A space cubic curve has the following form:
f(u) = u
g(u) = u²
h(u) = u³
The following figure shows this curve in the range of [-1,1]. It is contained in the box (in white) defined by ( -1, -1, 0 ) and (1, 1, 1)
The circular helix is defined as follows:
f(u) = ( acos(u), asin(u), bu )
The following figure shows the curve in the range of 0 and 4*PI. The initial point is (a, 0, 0) and the endpoint is (a, 0, b). Note that this curve lies on the cylinder of radius a and axis the z-axis.