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Discrete curvature

Ivan Izmestiev

TL;DR

The paper surveys discrete curvature notions across curves, surfaces, and polyhedral manifolds, tying them to classical results such as the Theorema Egregium and Gauss–Bonnet theorems. It develops both smooth and polyhedral perspectives, including Steiner–Minkowski theory, tangent indicatrices, Crofton-type integral geometry, and intrinsic versus extrinsic viewpoints, while establishing discrete analogues of fundamental identities. It also clarifies how discrete curvatures converge to their smooth counterparts under approximation and extends the framework to higher dimensions via Lipschitz–Killing curvatures and the Chern–Gauss–Bonnet theory. The work highlights a cohesive bridge between discrete and smooth geometry, with Regge-type formulations for polyhedral manifolds and a robust intrinsic–extrinsic calculus applicable to higher-dimensional spaces.

Abstract

The combination of words ``discrete curvature'' is only an apparent contradiction. In this survey we describe curvature notions associated with polygons, polyhedral surfaces, and with abstract polyhedral manifolds. Several theorems about the discrete curvature are stated that repeat literally classical theorems of differential and Riemannian geometry: Theorema Egregium, Gauss--Bonnet theorem, and the Chern--Gauss--Bonnet theorem among the others. Some convergence results are also mentioned: under certain assumptions the discrete curvature tends to the smooth curvature as a smooth object is approximated by polyhedral ones.

Discrete curvature

TL;DR

The paper surveys discrete curvature notions across curves, surfaces, and polyhedral manifolds, tying them to classical results such as the Theorema Egregium and Gauss–Bonnet theorems. It develops both smooth and polyhedral perspectives, including Steiner–Minkowski theory, tangent indicatrices, Crofton-type integral geometry, and intrinsic versus extrinsic viewpoints, while establishing discrete analogues of fundamental identities. It also clarifies how discrete curvatures converge to their smooth counterparts under approximation and extends the framework to higher dimensions via Lipschitz–Killing curvatures and the Chern–Gauss–Bonnet theory. The work highlights a cohesive bridge between discrete and smooth geometry, with Regge-type formulations for polyhedral manifolds and a robust intrinsic–extrinsic calculus applicable to higher-dimensional spaces.

Abstract

The combination of words ``discrete curvature'' is only an apparent contradiction. In this survey we describe curvature notions associated with polygons, polyhedral surfaces, and with abstract polyhedral manifolds. Several theorems about the discrete curvature are stated that repeat literally classical theorems of differential and Riemannian geometry: Theorema Egregium, Gauss--Bonnet theorem, and the Chern--Gauss--Bonnet theorem among the others. Some convergence results are also mentioned: under certain assumptions the discrete curvature tends to the smooth curvature as a smooth object is approximated by polyhedral ones.

Paper Structure

This paper contains 17 sections, 24 theorems, 73 equations, 9 figures.

Table of Contents

  1. Curvature of curves and polygons
  2. The total curvature of planar curves
  3. The total curvature of spatial curves
  4. The generalized total curvature
  5. Curvature of surfaces
  6. Surfaces
  7. Steiner formula and Minkowski theory
  8. The total mean curvature and integral geometry
  9. The total Gaussian curvature and the extrinsic Gauss-Bonnet theorem
  10. Theorema Egregium and the intrinsic Gauss--Bonnet theorem
  11. Surfaces of bounded total curvature
  12. Higher dimensions
  13. Curvature of Riemannian and polyhedral manifolds
  14. Riemannian manifolds and the total scalar curvature
  15. The total scalar curvature of polyhedral manifolds
  16. ...and 2 more sections

Key Result

Theorem 1.1

The total signed curvature of a closed plane curve is an integer multiple of $2\pi$:

Figures (9)

  • Figure 1: The signed curvature of curves and polygons.
  • Figure 2: A smooth curve with the turning number $\pm 2$ and a polygon with the turning number $0$.
  • Figure 3: The tangent indicatrix of a space polygon.
  • Figure 4: Determining the probability that a random great circle intersects a given arc.
  • Figure 5: Decomposition of the $r$-neighborhood of a convex polyhedron.
  • ...and 4 more figures

Theorems & Definitions (42)

  • Theorem 1.1
  • proof
  • Theorem 1.2: Fenchel Fen29
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Lemma 1.6: Crofton
  • proof
  • ...and 32 more