Gaussian and Mean Curvature

A sheet of paper rolls into a cylinder without a single wrinkle. It cannot cover a sphere without tearing. This is not a question of material stiffness: Gaussian curvature K of a cylinder is zero, of a sphere it is positive. Gauss proved in 1827 that K does not change under bending without stretching - the Theorema Egregium. Cartographers have known since then: a world map without distortion cannot be drawn. GPS knows it too: without the general relativistic curvature correction, navigation error would be 11 km per day.

• **Manifold learning:** UMAP, t-SNE, Isomap all operate on the curvature of the data space; the projection to 2D distorts exactly what K prevents from being preserved
• **3D smoothing:** mean curvature flow removes noise from LiDAR and MRI scans while preserving geometric features
• **Computer graphics:** adaptive mesh tessellation allocates triangles by curvature - sparse where K ~ 0, dense where curvature is large
• **GPS and general relativity:** Earth's spacetime is slightly curved - curvature correction is mandatory in onboard navigation software

Предварительные знания

• The Second Fundamental Form

First and Second Fundamental Forms

**Two questions about a surface.** First: how are distances and angles measured without leaving it? Second: how does the surface bend relative to the ambient space? Each question has its own form - first and second. Together they encode all of the geometry.

The **first fundamental form** I = E du^2 + 2F du dv + G dv^2 is the metric tensor of the surface. Coefficients: E = r_u·r_u, F = r_u·r_v, G = r_v·r_v. Arc lengths, angles between curves, areas of regions - all come from I. This is intrinsic geometry: a bug living on the surface, unaware of the third dimension, works exclusively with I.

The **second fundamental form** II = L du^2 + 2M du dv + N dv^2 measures how the surface deviates from its tangent plane. Coefficients: L = r_uu·n, M = r_uv·n, N = r_vv·n, where n is the unit normal. This is extrinsic geometry - visible from the ambient R^3. Manifold learning algorithms (UMAP, t-SNE, Isomap) navigate exactly this dichotomy: intrinsic distances versus extrinsic coordinates.

**Shape operator** S = I^(-1) * II links both forms. Principal curvatures κ₁, κ₂ are eigenvalues of S. Key formulas: K = (LN - M^2)/(EG - F^2), H = (EN - 2FM + GL)/(2(EG - F^2)).

Principal Curvatures κ₁ and κ₂

**Pizza confirms the theorem.** A slice of pizza is flat: K = 0. When folded along a radius, κ₁ is created along the fold line. But K = κ₁κ₂ must stay zero (Theorema Egregium - K is invariant under bending without stretching). Therefore κ₂ = 0 perpendicular to the fold - and the crust does not droop. This is not engineering intuition; it is mathematics.

The **normal curvature** in direction θ: κ(θ) = (L cos^2(θ) + 2M cos(θ)sin(θ) + N sin^2(θ)) / (E cos^2(θ) + 2F cos(θ)sin(θ) + G sin^2(θ)). It is the curvature of the normal section - the curve cut by the plane through the normal and direction θ.

**Principal curvatures** κ₁ and κ₂ are the maximum and minimum of normal curvature over all directions, achieved along mutually perpendicular **principal directions**. Euler's theorem: κ(θ) = κ₁ cos^2(θ) + κ₂ sin^2(θ) - all information is encoded in two numbers. In computer graphics, adaptive mesh tessellation places more triangles where κ₁ and κ₂ are large, fewer where they are near zero.

**Euler's theorem:** κ(θ) = κ₁ cos^2(θ) + κ₂ sin^2(θ), where θ is the angle from the first principal direction. All normal curvature information is encoded in just two numbers κ₁ and κ₂.

Principal directions are undefined at **umbilic points** where κ₁ = κ₂. Every point of a sphere is umbilic - there is no preferred bending direction. On a torus, umbilic points form two circles.

Gaussian Curvature K and Mean Curvature H

**Gottingen, 1827.** Gauss proves the Theorema Egregium - the "remarkable theorem". Gaussian curvature K is an intrinsic invariant: a creature living on the surface can compute K by measuring only distances, without ever leaving it. Eighty-eight years later Einstein uses exactly this for general relativity: spacetime signals its own curvature from within.

**Gaussian curvature** K = κ₁ * κ₂ = (LN - M^2)/(EG - F^2). K > 0 at elliptic points (bowl or dome), K < 0 at hyperbolic points (saddle), K = 0 at parabolic points (cylinder, cone). GPS satellites must correct for the curvature of Earth's spacetime - without the general relativistic correction, navigation error would accumulate at 11 km per day.

**Mean curvature** H = (κ₁ + κ₂)/2 = (EN - 2FM + GL)/(2(EG - F^2)). Physically, H governs the pressure jump across a soap film (Laplace law: ΔP = 2γH). Its sign depends on the normal orientation. In manifold learning (UMAP, t-SNE, Isomap), the curvature of the data manifold determines the distortions introduced when projecting to 2D.

Pringles chips are shaped as hyperbolic paraboloids (K < 0) deliberately: negative Gaussian curvature makes the chip structurally far stiffer than a flat wafer of the same thickness. The same principle governs shell roof architecture, saddle-shaped bridges, and helmet design.

Minimal Surfaces and Applications

**A soap film solves an optimization problem.** Stretched over a wire boundary, it takes the shape of least area for the given constraints. This is not a physical accident - it is a mathematical minimum: H = 0 everywhere. The catenoid (surface of revolution of the catenary) is minimal. Architects copy this geometry in shell coverings, where minimizing material while maintaining load capacity is the identical task.

**Mean curvature flow:** dr/dt = H * n. Each point moves with speed proportional to H along the normal. Sharp features with large |H| disappear first; smooth regions (H ~ 0) barely move. In 3D scanning (MRI, LiDAR) this flow is the standard denoising algorithm: it removes high-frequency noise while preserving the large-scale shape.

**Practical uses:** mean curvature flow removes noise from LiDAR/MRI scans, curvature descriptors power shape recognition in computer vision, and 3D printing support structures are optimized via the minimal surface condition H = 0 to minimize material.

Key Ideas

• **First form** I gives the surface metric (intrinsic geometry); **second form** II encodes bending in R^3 (extrinsic geometry)
• **Principal curvatures** κ₁, κ₂ are eigenvalues of S = I^(-1)·II, linked by Euler's theorem κ(θ) = κ₁ cos^2(θ) + κ₂ sin^2(θ)
• **K = κ₁κ₂** (Theorema Egregium): K > 0 elliptic, K < 0 hyperbolic, K = 0 parabolic; K is invariant under bending without stretching
• **H = (κ₁ + κ₂)/2**: H = 0 defines minimal surfaces; mean curvature flow smooths meshes by attacking the highest-H regions first

Related Topics

Surface curvature ties together all major areas of differential geometry:

• Second Fundamental Form — dg-03 introduces the shape operator S; here K and H are extracted from its eigenvalues
• Theorema Egregium — K turns out to be intrinsic - computable from I alone, without II
• Riemann Curvature Tensor — K generalizes to the full curvature tensor of Riemannian manifolds in any dimension

Вопросы для размышления

• The catenoid (surface of revolution of a catenary) is minimal with H = 0. Why does a soap film between two rings take exactly this shape and not a cylinder?
• Brain folds are mostly hyperbolic (K < 0). What does this reveal about the strategy for fitting a large cortex into a confined skull?
• Mean curvature flow is used to study loss surface geometry in ML. If a loss function has sharp minima (high curvature), what does this imply about the model's generalization?

Связанные уроки

• dg-03 — Shape operator S is the foundation; K and H are its eigenvalues
• dg-05 — Theorema Egregium: K is intrinsic, computable from the first form alone
• dg-10 — Riemann curvature tensor generalizes K to any-dimensional manifolds
• calc-15-convergence — Area functional and variational arguments use the same limit language
• alg-01 — Principal curvatures are eigenvalues - the same mechanism as PCA
• la-13-eigenvectors