Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
Harold "Hal" Abelson is a professor of electrical engineering and computer science at the Massachusetts Institute of Technology (MIT), a fellow of the Institute of Electrical and Electronics Engineers (IEEE), and a founding director of both Creative Commons and the Free Software Foundation.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial 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, polygons, and so forth. 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.
In computational geometry, the Theta graph, or -graph, is a type of geometric spanner similar to a Yao graph. The basic method of construction involves partitioning the space around each vertex into a set of cones, which themselves partition the remaining vertices of the graph. Like Yao Graphs, a -graph contains at most one edge per cone; where they differ is how that edge is selected. Whereas Yao Graphs will select the nearest vertex according to the metric space of the graph, the -graph defines a fixed ray contained within each cone and selects the nearest neighbor with respect to orthogonal projections to that ray. The resulting graph exhibits several good spanner properties.
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI, depending on the complexity of the domain and the type of mesh desired. The goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine in areas that are important for the subsequent calculations.
Prof. Athanasios K. Tsakalidis is a Greek computer scientist, a professor at the Graphics, Multimedia and GIS Laboratory, Computer Engineering and Informatics Department (CEID), University of Patras, Greece.
David Arthur Eppstein is an American computer scientist and mathematician. He is a Chancellor's Professor of computer science at the University of California, Irvine. He is known for his work in computational geometry, graph algorithms, and recreational mathematics. In 2011, he was named an ACM Fellow.
A geometric spanner or a t-spanner graph or a t-spanner was initially introduced as a weighted graph over a set of points as its vertices for which there is a t-path between any pair of vertices for a fixed parameter t. A t-path is defined as a path through the graph with weight at most t times the spatial distance between its endpoints. The parameter t is called the stretch factor or dilation factor of the spanner.
Godfried Theodore Patrick Toussaint was a Canadian Computer Scientist, a Professor of Computer Science, and the Head of the Computer Science Program at New York University Abu Dhabi (NYUAD) in Abu Dhabi, United Arab Emirates. He is considered to be the father of computational geometry in Canada. He did research on various aspects of computational geometry, discrete geometry, and their applications: pattern recognition, motion planning, visualization, knot theory, linkage (mechanical) reconfiguration, the art gallery problem, polygon triangulation, the largest empty circle problem, unimodality, and others. Other interests included meander (art), compass and straightedge constructions, instance-based learning, music information retrieval, and computational music theory.
Planarity is a puzzle computer game by John Tantalo, based on a concept by Mary Radcliffe at Western Michigan University. The name comes from the concept of planar graphs in graph theory; these are graphs that can be embedded in the Euclidean plane so that no edges intersect. By Fáry's theorem, if a graph is planar, it can be drawn without crossings so that all of its edges are straight line segments. In the planarity game, the player is presented with a circular layout of a planar graph, with all the vertices placed on a single circle and with many crossings. The goal for the player is to eliminate all of the crossings and construct a straight-line embedding of the graph by moving the vertices one by one into better positions.
Kurt Mehlhorn is a German theoretical computer scientist. He has been a vice president of the Max Planck Society and is director of the Max Planck Institute for Computer Science.
János Pach is a mathematician and computer scientist working in the fields of combinatorics and discrete and computational geometry.
Roberto Tamassia is an American Italian computer scientist, the Plastech Professor of Computer Science at Brown University, and served as the chair of the Brown Computer Science department from 2007 to 2014. His research specialty is in the design and analysis of algorithms for graph drawing, computational geometry, and computer security; he is also the author of several textbooks.
A Reeb graph is a mathematical object reflecting the evolution of the level sets of a real-valued function on a manifold. According to a similar concept was introduced by G.M. Adelson-Velskii and A.S. Kronrod and applied to analysis of Hilbert's thirteenth problem. Proposed by G. Reeb as a tool in Morse theory, Reeb graphs are the natural tool to study multivalued functional relationships between 2D scalar fields , , and arising from the conditions and , because these relationships are single-valued when restricted to a region associated with an individual edge of the Reeb graph. This general principle was first used to study neutral surfaces in oceanography.
Emmerich (Emo) Welzl is a computer scientist known for his research in computational geometry. He is a professor in the Institute for Theoretical Computer Science at ETH Zurich in Switzerland.
Ferran Hurtado Díaz was a Spanish mathematician and computer scientist known for his research in computational geometry.
