How Surfaces Curve

Differential Geometry · Axiom Academy

Discover the geometry of curved surfaces through interactive exploration. Step 1: Bending in Different Ways Let's explore how a simple flat surface can bend in different directions. Adjust the curvature sliders to see how the surface responds. Click on each surface type to see how different curvature combinations create distinct shapes. Positive (Bowl): Both principal curvatures have the same sign Negative (Saddle): Principal curvatures have opposite signs Zero (Cylinder): At least one principal curvature is zero Rotate the surface to see how curvature reveals important geometric properties that aren't obvious at first glance. Physics: Curvature describes spacetime in General Relativity Engineering: Stress analysis on curved structures Computer Graphics: Realistic surface rendering and lighting Biology: Understanding protein folding and cell membranes Step 4: Two Ways to Measure Curvature At each point on a surface, there are two important curvature measures. Explore how they relate to surface shape. K = κ₁ × κ₂ Product of principal curvatures Intrinsic property H = (κ₁ + κ₂) / 2 Average of principal curvatures Extrinsic property Gaussian Curvature (K): Tells you about the surface's intrinsic geometry. A cylinder and a flat sheet have K = 0, which is why you can roll paper into a cylinder without stretching! Mean Curvature (H): Describes how the surface bends in 3D space. Minimal surfaces (like soap films) have H = 0 everywhere.

This is the written version of the interactive lesson above. See the full Differential Geometry course.