Rational Curves

Parametric representations using polynomials are simply not powerful enough, because many curves can not be obtained this way. See one of the problems of this unit. One way to overcome this is to use homogeneous coordinates. For example, a curve in space (resp., plane) is represented with four (resp., three) functions rather than three as follows:

Space curve: F(u) = ( x(u), y(u), z(u), w(u) )
Plane curve : F(u) = ( x(u), y(u), w(u) )
where u is a parameter in some closed interval [a,b]. Converting this curve to their conventional form yields the following:
Space curve: f(u) = ( x(u) / w(u), y(u) / w(u), z(u) / w(u) )
Plane curve : f(u) = ( x(u) / w(u), y(u) / w(u) )
Obviously, if w(u) = 1, a constant function, the homogeneous forms reduce to conventional forms.

A parametric curve in homogeneous form is referred to as a rational curve. To make a distinction, we shall call a curve in polynomial form a polynomial curve.

An Example

Let us use an example to illustrate the power of the rational form. Consider a second degree parametric form:
x = f(u) = au2 + bu + c
y = g(u) = pu2 + qu + r
Let the domain of u be the real line. Now, let u run from negative infinity to positive infinity. The values of x and y will both go to infinity, positive or negative. In other words, the curve described by the above parametric form contains a point at infinity. Since all points of a circle are in finite range, it is impossible to obtain a circle from the above parametric form. So, this shows a weakness of parametric form.

Rational Forms of Common Curves

First, let us find the rational form of a circle.

The above shows a circle with center at the origin and radius 1. Let u be a parameter for this circle. For convenience purpose, we use the x-axis for the value of u and as a result for any particular value of u there is a point (u,0) on the x-axis. Join a line between (u,0) and the south pole of the circle (0,-1). Since a line intersects a circle in two points, let the other point which is not the south pole be (x(u), y(u)). As u moves on the x-axis, its corresponding point moves on the circle. Any finite u has such a corresponding point. The infinite u corresponds to the south pole.

The line joining the south pole and (u,0) is x = uy + u = 0. The circle equation is x2 + y2 = 1. Plugging the line equation into the circle equation and solving for y yields two roots for y. One of them must be y = -1, since this line passes through the south pole. The other root is y = (1 - u2) / (1 + u2). Plugging this y into the line equation yields x = (2 u) / (1 + u2). Therefore, for each u, the corresponding point on the circle is

x = 2 u / (1 + u2)
y = (1 - u2) / (1 + u2)
As a result, a circle has a rational parameterization. When u moves to infinity, x approaches 0 while y approaches -1 (use L'Hospital's rule).

In fact, this computation tells us more. This circle has a trigonometric parametric form ( cos(t), sin(t) ), where t is in the range of 0 and 2PI. Therefore, a point on the circle can have two different representation, but their values are equal. That is, ( cos(t), sin(t) ) = ( (2 u) / (1 + u2), (1 - u2) / (1 + u2) ) for some u. Therefore, trigonometric functions cos(t) and sin(t) can be parameterized as follows:

cos(t) = 2 u / (1 + u2)
sin(t) = (1 - u2) / (1 + u2)

With parameterization for cos(t) and sin(t) in hand, you can easily find a rational parameterization for an ellipse and a hyperbola. Let us do the hyperbola part and leave the ellipse as an exercise. A hyperbola in its normal form has an equation as follows:

x2/a2 - y2/b2 = 1
where a and b are the semi-major and semi-minor axes lengths. It is easy to verify that the following is a correct parameterization for the hyperbola:
x = a sec(t)
y = b tan(t)
Since tan(t) = sin(t) / cos(t) and sec(t) = 1 / cos(t), plugging the rational parameterization for sin(t) and cos(t) into the above equations will convert them from a trigonometric parametric form to a rational one as follows:
x = a (1 + u2) / (2u)
y = b (1 - u2) / (2u)

Uniformization Theorems

You have seen the way of converting a polynomial parametric form to a polynomial p(x, y) = 0 in a previous example. Rational forms can also be converted to polynomials. The trick is by eliminating the parameter u from all parametric equations. It is tedious; but it can be done. In fact, with elimination theory, one can convert a polynomial or rational parametric form to an equivalent implicit polynomial in two variables. This is called implicitization.

The question is: given a curve in an implicit polynomial form, is it possible to find a parametric form, polynomial or rational, that describes the same curve? Unfortunately, the answer is negative. That is, there exists curves which do not have any polynomial or rational parametric forms (e.g., the elliptic curve y2 = x3 + ax + c). This is the result of uniformization theorem.

Due to this fact, implicit polynomials can describe more curves than polynomial and rational parametric forms. Why don't we use implicit polynomials? The reason is simple. Parametric curves are easy to use. While implicit polynomial curves are powerful and can describe more curves, many of its important properties are difficult to compute and implicit curves are in general difficult to use. We shall cover implicit curves and surfaces later in this class.