Discrete differential geometry

Geometry
Projecting a sphere to a plane
Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry
Concepts Features Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry
Zero-dimensional Point
One-dimensional Line segment ray Curve Length
Two-dimensional Surface Plane Area Polygon Triangle Altitude Hypotenuse Pythagorean theorem Quadrilateral Parallelogram Square Rectangle Rhombus Rhomboid Trapezoid Kite Circle Diameter Circumference Area
Three-dimensional Volume Polyhedron Platonic Solid Tetrahedron cuboid Cube Octahedron Dodecahedron Icosahedron Pyramid Solid of revolution Sphere Cylinder Cone
Four-/other-dimensional Tesseract Hypersphere
Geometers
by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers
by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Huygens Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Chern Present day Atiyah Gromov
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Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, geometry processing and topological combinatorics.

Discrete differential geometry (DDG) aims not merely to discretize objects or equations, but to discretize the entire theory of classical differential geometry. In this context, classical differential geometry is expected to emerge as a limit of refinement of the discretization. Furthermore, in the process of refinement, DDG places an emphasis on the so-called “mimetic” viewpoint, that is, whether the essential properties of the system from the smooth setting are exactly preserved regardless of the size of the mesh elements.

Generally, for a given smooth geometry, one can suggest many different discretizations with the same continuous limit. In other words, there is no single “correct” way to discretize a given geometric quantity; rather, there are various different ways, each suited to specific purposes. Therefore, it is necessary to choose the appropriate discretization suited to your purpose (the so-called “game” of DDG).

Integrable discretization is a systematic discretization based on integrable systems, which originates from the study of localized waves called solitons in mathematical physics. Integrable discretization gives an efficient (and nearly algorithmic) method to discretize a theory (not just its objects). Since curves and surfaces consistent with discrete integrable systems are in a sense “good,” integrable discretization has been applied in fields such as industrial design, computational architecture, and others.

Introductory text: A. I. Bobenko, Y. B. Suris, “Discrete Differential Deometry. Consistency as Integrability.” arXiv, 2005, arXiv:math/0504358 [math.DG]

In contrast to the purely geometric perspective, differential operators provide a very different point of view. This dual perspective enriches understanding on both sides, and leads to the development of practical algorithms for working with real-world geometric data. Discrete Laplacians ${\displaystyle \Delta }$ are defined using discrete exterior calculus (DEC). In general, the smooth Laplacian has variety of discretizations depending on the definition of inner product (the “game” of DDG). However, it is known that certain types of meshes allow for “perfect” discrete Laplacians, offering a connection between geometry and (discrete) differential operators.

Introductory text: K. Crane, “Discrete Differential Geometry: An Applied Introduction,” 2025.

The research subject of the discrete differential geometry of n-simplices (DDGNS) is “polygonal lines” consisting of n-simplices, i.e., n-dimensional objects obtained by connecting n-simplices one by one via their common faces in Rⁿ. DDGNS focuses primarily on “quantization” rather than “discretization” of classical differential geometry. Just as classical mechanics does not appear as a smooth limit of quantum mechanics, classical differential geometry does not appear as a smooth limit of DDGNS. Currently, DDGNS is applied in the field of protein science.

Introductory text: N. Morikawa, “A Novel Mathematical Model of Protein Interactions from the Perspective of Electron Delocalization,” arXiv:2509.18882 [q-bio.BM]

• Discrete Laplace operator
• Discrete exterior calculus
• Discrete Morse theory
• Topological combinatorics
• Spectral shape analysis
• Analysis on fractals
• Discrete calculus

• Discrete differential geometry Forum
• Keenan Crane; Max Wardetzky (November 2017). "A Glimpse into Discrete Differential Geometry". Notices of the American Mathematical Society. 64 (10): 1153–1159. doi:10.1090/noti1578.
• Alexander I. Bobenko; Peter Schröder; John M. Sullivan; Günter M. Ziegler (2008). Discrete differential geometry. Birkhäuser Verlag AG. ISBN 978-3-7643-8620-7.
• Alexander I. Bobenko, Yuri B. Suris (2008), "Discrete Differential Geometry", American Mathematical Society, ISBN 978-0-8218-4700-8

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