Discrete Differential Geometry of Curves and Surfaces

Introduction

Discrete Differential Geometry (DDG) extends concepts from classical differential geometry to discrete settings, where curves and surfaces are represented by polygonal chains or triangle meshes. This field has grown significantly in recent decades due to applications in computer graphics, geometric modeling, computational architecture, and other fields working with digital geometric representations.

Unlike classical differential geometry dealing with smooth, infinitely differentiable curves and surfaces, DDG focuses on structures composed of vertices, edges, and faces. Despite this discrete nature, DDG preserves many important properties and theorems from its continuous counterpart through clever adaptations that account for the discrete representation.

The beauty of DDG lies in bridging theoretical mathematics and practical computational systems, providing rigorous foundations for algorithms that manipulate geometric data while ensuring important geometric invariants are preserved or correctly approximated.

Discrete Curves

In DDG, a curve is typically represented as a polygonal chain - a sequence of points (vertices) connected by straight line segments (edges). This discrete representation contrasts with continuous parametric curves but enables computation of analogous geometric quantities.

Discrete Curvature

For curves, curvature measures how much a curve deviates from being straight. In a discrete setting, curvature at a vertex is typically defined using the turning angle. For three consecutive vertices A, B, and C, the discrete curvature at vertex B is given by the angle formed by line segments BA and BC:

This definition has desirable properties: it vanishes for collinear points, is invariant under rigid transformations, and converges to continuous curvature with dense sampling of smooth curves.

Discrete Torsion

For three-dimensional curves, torsion measures twisting out of the plane of curvature. The discrete analog uses four consecutive vertices. For vertices A, B, C, and D, the discrete torsion relates to the dihedral angle between planes formed by {A, B, C} and {B, C, D}.

These discrete operators form the foundation for discrete curve algorithms, including curve fairing, approximation, and animation.

Discrete Surfaces

Discrete surfaces are typically represented as triangle meshes - collections of triangular faces connecting vertices in three-dimensional space. These meshes form the backbone of 3D computer graphics and modeling. DDG extends concepts from smooth surface theory to these polygonal representations.

Discrete Gaussian Curvature

Gaussian curvature has a natural discrete counterpart. At a vertex of a triangle mesh, discrete Gaussian curvature K is defined as an angle defect:

where i are the angles at the vertex in each incident triangle. This captures intrinsic curvature - measuring how much the sum of angles around a vertex deviates from the 2 expected for a flat surface. For convex vertices, curvature is positive; for saddle-like vertices, it's negative; and for flat vertices, it's zero.

A remarkable property is that the discrete Gauss-Bonnet theorem holds. For a closed discrete surface S, the sum of discrete Gaussian curvatures at all vertices equals 2 times the Euler characteristic:

Discrete Mean Curvature

Mean curvature, relating to surface energy and minimal surfaces, has discrete analogs using the cotangent formula. For a vertex V in a triangle mesh, the discrete mean curvature normal vector H is:

where A is the Voronoi area around vertex V, and are angles opposite to edge V-Ni in adjacent triangles, and the sum runs over all neighboring vertices Ni.

This discrete mean curvature is the gradient of the Dirichlet energy on the mesh and relates to the Laplace-Beltrami operator, central to many mesh processing algorithms.

Discrete Laplace-Beltrami Operator

The Laplace-Beltrami operator, a fundamental differential operator on smooth surfaces, has a discrete counterpart essential for geometry processing. For a function f defined on mesh vertices, the Laplacian at a vertex V is:

where A is the Voronoi area, and are angles opposite to edge V-Ni in adjacent triangles, and the sum runs over all neighboring vertices Ni.

Applications

Discrete differential geometry has numerous applications across various fields, showcasing its practical significance alongside theoretical elegance.

Computer Graphics and Animation

In computer graphics, discrete differential operators are used for mesh smoothing, deformation, and editing. The discrete Laplace-Beltrami operator is fundamental to algorithms for implicit fairing, which smooths a surface while preserving its shape. In animation, discrete curve and surface animations leverage geometric invariants for natural-looking motions.

Geometric Modeling

Geometric modeling systems use DDG concepts for parameterization, remeshing, and subdivision surfaces. Parameterization algorithms, mapping 3D surfaces to 2D planes, often rely on discrete conformal mapping techniques. Remeshing algorithms create meshes with desirable properties by controlling discrete curvature distributions.

Computational Architecture

Architects employ DDG to design complex structures like freeform roofs and facades. Concepts such as discrete minimal surfaces, developable surfaces, and discrete differential operators help create structurally efficient and aesthetically pleasing designs that can be realized in actual buildings.

Medical Imaging

In medical imaging, DDG aids in analyzing anatomical structures from 3D data. Techniques based on discrete curvature, geodesics, and shape descriptors help with segmentation, registration, and morphological comparison of biological shapes.

Conclusion

Discrete differential geometry stands at the intersection of pure mathematics and practical computation. By extending differential geometric concepts to discrete representations, it provides theoretical foundations for numerous algorithms and applications that shape our digital world.

The elegance of DDG lies in preserving important properties from smooth differential geometry while adapting them to computational settings. The discrete Gaussian curvature, mean curvature, and Laplace-Beltrami operator maintain key theoretical properties while leading to efficient algorithms with wide-ranging applications.

As computational power increases and new applications emerge, DDG continues to evolve. Researchers constantly develop new discrete operators, explore connections to other mathematics branches, and find innovative applications from scientific computing to artistic design. The future promises further advances and sophisticated tools for working with digital representations of the geometric world.