Geometry on Curved Surfaces

The geometry that we learned in school is not the only geometry known. That geometry is called Euclidean geometry. It is valid only in spaces that have zero curvature ("flat spaces").

In curved spaces (like the sphere and the saddle-shaped surfaces in the adjacent frame), the geometry is called non-Euclidean geometry, and some of the rules of normal geometry are no longer valid. For example, on curved surfaces the sum of the angles of a triangle is not 180 degrees.

Animation: geometry on curved surfaces