Discrete Geometry

Discrete geometry plays an expanding role in the fields of shape modeling, image synthesis and image analysis. It deals with the structure and complexity of discrete geometric objects. Discrete geometry is interdisciplinary in nature and has relations to many other vital mathematical methods, such as algebraic geometry, topology, combinatorics and probability theory. It’s also the cutting edge of modern applications such as geographic information systems, mathematical programming, coding theory, solid modeling and computational structural biology.

Discrete geometry and combinatorial geometry are branches of geometry that studies combinational properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects such as points, lines, planes, circles, spheres, and polygons. The subject focuses on the combinatorial properties of these objects such as how they intersect one another or how they may be arranged to cover a larger object.

Discrete geometry shares common characteristics with convex geometry and computational geometry and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometry graph theory, toric geometry and combinatorial topology.

History

Modern discrete geometry has its origins in the late 19^th century although polyhedral and tessellations have been studied for many years by people such as Kepler and Cauchy. The early topics studied under this discipline include the density of circle packings by Thue, Projective configurations by Reye and Steinitz, the Geometry of numbers by Minkowski and Map colourings by Tait.

Convex geometry

Convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas of mathematics: computational theory, convex analysis, discrete geometry, and functional analysis, geometry of numbers, integral geometry, linear programming and probability theory.

Computational geometry

Computational geometry is a branch of computer science developed to study the algorithms which can be stated in terms of geometry.

Discrete differential geometry

Discrete geometry is the study of discrete counter parts of concepts in differential geometry. It is used in the study of computer graphics and topological combinations.

Geometric graph theory

A geometric graph is a graph in which the vertices or edges are associated with geometric objects or configurations. Geometric graph theory is a specialization of graph theory that studies geometric graphs.

Finite geometry

A finite geometry is any geometric system which has only a limited number of points.  A finite geometry can have any number of dimensions.

Questions:

• What is discrete geometry?
• What are the different types of geometry associated with discrete geometry?