Gaussian Curvature vs. Mean Curvature: Geometry’s Hidden Shapes
Curvature quantifies how a surface bends at a point, revealing intrinsic and extrinsic properties that shape everything from mathematical surfaces to physical systems. Two fundamental types—Gaussian curvature and mean curvature—offer complementary insights into a surface’s geometry. Gaussian curvature, an intrinsic measure tied to how angles and areas behave locally, captures topology: a sphere’s positive curvature, a saddle’s negative curvature, and flat planes’ zero curvature. Mean curvature, by contrast, reflects how a surface sits within ambient space, averaging principal curvatures and governing shape deformation under external forces. Together, these curvatures uncover hidden structure invisible to simple flatness or smoothness.
The Core Distinction: Intrinsic vs. Extrinsic CurvatureGaussian curvature is intrinsic—its value depends only on measurements made within the surface itself,