TABLE OF CONTENTS In this chapter, we will be firmly rooted in Euclidean geometry. The curves we are going to discuss all satisfy a property that makes some associated vectors be of unit length, for example, tangent vectors.
A well-known parametrization of the circle may be obtained as follows. Consider a unit circle with center at the origin of a Euclidean coordinate system. We can write the southeast quarter as a rational quadratic with control points
and standard weights 1, 
,1. We may also reparametrize this circle representation, using a reparametrization constant c. [1] The equation then becomes
| (10.1) |  |
with c = 1 for the familiar standard form. Let us try to arrange the parametrization such that the curve passes through n = [0,1] T, the north pole of the circle, for t = ?. This leads to
from which we deduce 
. We may then take the reparametrized B zier form (10.1) and rewrite it as follows:
| (10.2) |  |
Check that x 2( t) + y 2( t) = 1!
This parametrization is an example of the stereographic projection: the line through the north pole n and [ x( t), y( t)] T intersects the x-axis in x = t. This is seen by checking that the three points
are collinear.
Thus every point on the x-axis has a unique image point on the circle and...
