The Gauss Map

The Gauss map of an oriented surface in Euclidean 3-space associates to each point on the surface its unit normal vector, considered as a point on the unit two-sphere. A pattern or texture on the two-sphere can be used to paint a surface by using the Gauss map to pull back the pattern to the surface. This is what is done in the two sequences below for the Costa surface. At all three of the ends of the Costa surface, the Gauss map is vertical. The patterns taken together illustrate the fact that at the catenoidal ends (the top and bottom ends) the Gauss map is regular, while at the flat middle end, the Gauss map has multiplicity three: note the three-to-one banding in the longitudinal picture.


Figure 1: Longitudinal sections on the sphere pulled back to the Costa Surface



Figure 2: Latitudinal sections on the sphere pulled back to the Costa Surface