Flächenkrümmung

Flächenkrümmung, or surface curvature, is a fundamental concept in differential geometry that describes how a surface bends in three-dimensional space. It quantifies the intrinsic deviation of a surface from being flat. There are several ways to measure curvature, each providing different insights.

The most common measures are the Gaussian curvature and the mean curvature. Gaussian curvature, denoted by K, is the product of the principal curvatures of the surface at a given point. These principal curvatures represent the maximum and minimum normal curvatures at that point. A positive Gaussian curvature indicates a region that curves away from the tangent plane in all directions, like a sphere or an ellipsoid. A negative Gaussian curvature signifies a region that curves in opposite directions, such as a saddle shape. Zero Gaussian curvature means the surface is flat in at least one direction, like a cylinder or a cone.

Mean curvature, denoted by H, is the average of the principal curvatures. It is related to the shape of a soap film or a membrane stretched over a boundary. Surfaces with zero mean curvature are called minimal surfaces.

These curvature measures are intrinsic properties, meaning they can be determined by measurements made entirely within the surface itself, without reference to the embedding space. This is a crucial distinction from extrinsic measures of curvature. Flächenkrümmung plays a vital role in fields such as general relativity, where it describes the curvature of spacetime, and in computer graphics and engineering for surface modeling and analysis.