Normal and Gaussian Curvature of Surfaces Explained

Normal curvature measures how sharply a surface bends in one chosen direction: slice the surface with the plane containing the direction and the surface normal, and take the curvature of the slice curve. Gaussian curvature multiplies the two extreme values, the principal curvatures, into one number K = kappa_1 kappa_2.

Gaussian curvature is profound because of Gauss's Theorema Egregium: K is computable from the first fundamental form alone, so it is intrinsic. A sphere cannot be flattened without distortion precisely because it has K > 0 while the plane has K = 0, which is why every flat world map distorts something.

Key Concepts

• Normal curvature: kappa_n = II(v,v)/I(v,v), the signed bending in a tangent direction.
• Principal curvatures: kappa_1, kappa_2, the maximum and minimum normal curvatures, in perpendicular directions.
• Gaussian curvature: K = kappa_1 kappa_2, units 1/length^2; positive is bowl-like, negative saddle-like, zero flat in some direction.
• Theorema Egregium: bending without stretching never changes K.
• Point types: elliptic (K>0), hyperbolic (K negative), parabolic (K=0, not both principal curvatures zero), planar.

Formulas

• K from the forms: K = (eg - f^2)/(EG - F^2).
• Sphere radius R: K = 1/R^2 everywhere; normal curvatures have magnitude 1/R.
• Plane and cylinder: K = 0, since a principal curvature vanishes.
• Needs: regular parametrisation so EG - F^2 > 0; K itself is independent of the normal's sign.

Worked Example

Gaussian curvature of the sphere radius 2: X(theta,phi) = (2 sin theta cos phi, 2 sin theta sin phi, 2 cos theta).

1. E = 4, F = 0, G = 4 sin^2 theta; outward normal n = (sin theta cos phi, sin theta sin phi, cos theta).
2. e = X_theta_theta·n = -2, f = 0, g = X_phi_phi·n = -2 sin^2 theta.
3. K = (eg - f^2)/(EG - F^2) = (4 sin^2 theta)/(16 sin^2 theta) = 1/4.
4. Check: 1/R^2 = 1/4; normal curvatures have magnitude 1/2, the reciprocal of the radius.

Common Misconceptions

• K = 0 is not flatness. A cylinder has K = 0 yet looks curved; it only bends in one direction per point, like a rolled sheet.
• Gaussian curvature is intrinsic, "looking curved" is not: cylinder and plane share K = 0 and identical intrinsic geometry.
• K > 0 means same-sign principal curvatures, not identical shape; an egg and a sphere both qualify.

Real-World Applications

• Cartography: no perfect flat map of the positively curved Earth exists; every projection sacrifices something.
• Architecture: saddle roofs (K negative) and domes (K>0) carry loads very differently.
• Manufacturing: panels with K ≠ 0 cannot be made by bending flat sheet alone.

Normal curvature slices bending in one direction; Gaussian curvature K = kappa_1 kappa_2 fuses the extremes into one intrinsic number. The radius-2 sphere has K = 1/4 everywhere. Gauss's discovery that K is measurable from inside the surface is why flattening a sphere without distortion is impossible.