A Rational Parameterization of the Unit Circle
We’re all familiar with the usual trigonometric parameterization of the unit circle: Each point on is given by for some real . Less well-known is the parameterization of the unit circle by rational functions.
The line through the point with slope is given by . This line intersects the unit circle in one other point and as we vary we strike every point on the unit circle. Here’s an illustration for a few values of :
What are the coordinates of this point? Well, the point satisfies both and so we have
corresponds to the point . When we have
and we want the to be so that positive slopes correspond to the upper half of the circle as we illustrated above.
Therefore every point on the unit circle (other than ) is of the form
for some . As we have .
We now have two parameterizations of the unit circle. How are they connected? For every there is a such that
This calls to mind the tangent addition formula . This suggests and . The usual calculations show that this is correct: If we start with we get the desired and .