A problem in the Problems section addresses some of the minor flaws in the above calculation. Rational Forms of Common Curves 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. 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. On the other hand, any point on the circle that is not the south pole corresponds to a u on the x -axis. Plugging the line equation into the circle equation and solving for y yields two roots for y . As a result, a circle has a rational parameterization. In fact, this computation tells us more.