Gaussian & Mean Curvature

The two fundamental curvature measures with stunning color-coded visualizations

Gaussian & Mean Curvature

Two numbers sum up how a surface bends at a point: Gaussian curvature K tells bowl from saddle, and mean curvature H measures the average bend.

Gaussian Curvature Visualization

Pick a surface and read its colors: red where K is positive (bowl-like), blue where it is negative (saddle-like), white where it is zero. On the torus, compare the outer and inner rims.

Gaussian Curvature K
K < 0 (Saddle)
K = 0 (Flat/Cylindrical)
K > 0 (Bowl/Sphere)
Torus

K varies: positive outside, negative inside

K range: [0.000, 0.000]
Gaussian Curvature
K = k₁ · k₂
Product of principal curvatures
Go deeper:Gaussian Curvature K

Curvature quantifies how a surface bends at each point. There are two fundamental measures: Gaussian curvature K (the product of principal curvatures) and mean curvature H (their average).

These quantities reveal deep information about the geometry of a surface. Gaussian curvature is intrinsic — it can be measured by a 2D being living on the surface — while mean curvature describes how the surface bends in the surrounding 3D space.

K = k₁ · k₂
  • K > 0: Elliptic point (sphere-like, both principal curvatures same sign)
  • K < 0: Hyperbolic point (saddle-like, opposite signs)
  • K = 0: Parabolic point (at least one k = 0, like a cylinder)
Go deeper:Mean Curvature H
H = (k₁ + k₂) / 2
  • H = 0: Minimal surface (soap films, catenoid, helicoid)
  • H > 0: Surface curves toward normal direction
  • H < 0: Surface curves away from normal direction
Go deeper:Intrinsic vs Extrinsic Geometry

A remarkable fact: Gaussian curvature K is intrinsic — it depends only on measurements made within the surface itself. This is Gauss's "Theorema Egregium" (remarkable theorem).

In contrast, mean curvature H is extrinsic — it depends on how the surface sits in 3D space. You could bend a flat sheet of paper (K = 0 everywhere) into a cylinder without stretching it, but a 2D being on the paper couldn't detect the bending.

Key Takeaways

  • K = k₁k₂ classifies points as elliptic (K > 0), hyperbolic (K < 0), or parabolic (K = 0).
  • H = (k₁ + k₂)/2 measures average bending; H = 0 defines minimal surfaces.
  • K is intrinsic: a 2D being can measure it without leaving the surface.
  • H is extrinsic: it changes when you bend the surface in space.

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