The Theorema Egregium ('Remarkable Theorem') is an important theorem of Gauss concerning the curvature of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface, that is, it does not depend on how the surface might be imbedded in (3-dimensional) space.
Gauss presented the theorem this way (translated from Latin):
- Thus the formula of the preceeding article leads itself to the remarkable
- Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.
- The Gaussian curvature of a surface is invariant under local isometries.
You can't bend a piece of paper onto a sphere (more formally, the plane and the 2-sphere are not locally isometric). The follows immediately from the fact that the plane has Gaussian curvature 0 (at all points) while no point on a sphere always has Gaussian curvature 0. (It is, however, possible to prove this special case more directly.)
Corresponding points on the catenoid and the helicoid (two very different-looking surfaces) have the same Gaussian curvature. (The two surfaces are locally isometric.)Some simple applications