Surface Curvature Examples

Differential Geometry · Axiom Academy

EXAMPLE Surface Curvature Examples Computing Gaussian and mean curvature for classic surfaces Excellent work! You've explored curvature computations for four fundamental surfaces: Sphere: Constant positive Gaussian curvature K = 1/R² reflects uniform bending in all directions. Both principal curvatures equal 1/R. Cylinder: Zero Gaussian curvature (K = 0) indicates it can be "unrolled" to a plane without distortion. It curves in only one direction. Saddle Surface: Negative Gaussian curvature (K < 0) comes from opposite-sign principal curvatures, creating the characteristic saddle shape. Paraboloid: Positive Gaussian curvature like a sphere, but varies with position. At the vertex, it approximates a sphere of radius a. These examples illustrate how Gaussian curvature K and mean curvature H capture the intrinsic and extrinsic geometry of surfaces. Practice computing these for other surfaces to deepen your understanding!

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