Discrete differential geometry is the study of discrete counterparts of notions in
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
. Instead of smooth curves and surfaces, there are
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
s,
meshes
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, exp ...
, and
simplicial complexes. It is used in the study of
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
,
geometry processing
Geometry processing, or mesh processing, is an area of research that uses concepts from applied mathematics, computer science and engineering to design efficient algorithms for the acquisition, reconstruction, analysis, manipulation, simulatio ...
and
topological combinatorics.
See also
*
Discrete Laplace operator
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertic ...
*
Discrete exterior calculus In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs and finite element meshes. DEC methods have proved to be very powerful in improving and analyzing finite element ...
*
Discrete Morse theory
*
Topological combinatorics
*
Spectral shape analysis Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, i ...
*
Abstract differential geometry
*
Analysis on fractals
*
Discrete calculus
References
Discrete differential geometry Forum*
*
* Alexander I. Bobenko, Yuri B. Suris (2008), "Discrete Differential Geometry", American Mathematical Society,
Differential geometry
Simplicial sets
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