Gaussian curvature is an intrinsic measure of curvature that depends solely on distances measured on a surface, independent of how the surface is embedded in space. It is calculated as the product of the principal curvatures at a given point, and it determines whether the surface is locally shaped like a sphere, a saddle, or a flat plane.

Gaussian Curvature

Gaussian curvature

intrinsic measure of curvature

distances measured on a surface

Surface embedding

Principal curvatures are the maximum and minimum values of the curvature of a surface at a given point, providing critical insight into the surface's local shape. They are essential in differential geometry for understanding the intrinsic and extrinsic properties of surfaces, influencing applications in computer graphics, physics, and engineering design.

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