The Gauss’s, Theorema Egregium

Abstract

To understand Gauss’s theory about the non-Euclidean geometry we have to reestablish some definitions of the coordinate system, and introduce the so-called Gaussian coordinates. We show here that the two points distance as a postulate can establish a metric geometry. If we can show the validity of this postulate on any surface than it has his geometry, and not necessarily Euclidean. Gauss showed in The Theorema Egregium that a surface might have such attributes. The different geometries of the regular surfaces written here are Euclidean, spherical, and hyperbolic. This theorem presented in 1827. (Based on the lectures of K. Lanczos: Department of Physical Sciences and Applied Mathematics, North Carolina State University, Raleigh, 1968.)The importance of this lecture is to make clear and understandable by using Gausss’s theorem how and why the physicians must use non-Euclidean geometry.