Constructions of Rational Surfaces in the Three-Sphere

The three-dimensional unit sphere is a quadratic hypersurface embedded in the four-dimensional Euclidean space. Both the Euclidean 3-space and the unit 3-sphere possess a natural representation in terms of Hamilton's quaternions. Then polynomial Bezier surfaces can be considered as parametric surfaces placed in the space of pure quaternions. Using stereographic projection we obtain a rational surface in the unit 3-sphere which is associated to any polynomial Bezier surface. These spherical surfaces are described in terms of the unit quaternions.