Then use the techniques of finding surface area from vector analysis. We can easily flatten the cylinder onto a plane and find its area to be 4 p r 2 . The area of a sphere of radius r is 4 p r 2 . This is called stereographic projection . Show that stereographic projection preserves the sizes of angles.
Stereographic projection is angle-preserving .
There are several approaches for exploring this problem. Using a purely geometric approach requires visualization but only very basic geometry. An analytic approach requires knowledge of the differential of a function from R 2 into R 3 . An angle at a point X on the sphere is determined by two great circles intersecting at X . Look at the two planes that are determined by the North Pole N and vectors tangent to the great circles at X .