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In geometry, a triangulation is a subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object into simplices. Triangulations of a three-dimensional volume would involve subdividing it into tetrahedra packed together.
In most instances, the triangles of a triangulation are required to meet edge-to-edge and vertex-to-vertex.
Different types of triangulations may be defined, depending both on what geometric object is to be subdivided and on how the subdivision is determined.
The concept of a triangulation may also be generalized somewhat to subdivisions into shapes related to triangles. In particular, a pseudotriangulation of a point set is a partition of the convex hull of the points into pseudotriangles—polygons that, like triangles, have exactly three convex vertices. As in point set triangulations, pseudotriangulations are required to have their vertices at the given input points.
In computational geometry, a Delaunay triangulation or Delone triangulation of a set of points in the plane subdivides their convex hull into triangles whose circumcircles do not contain any of the points. This maximizes the size of the smallest angle in any of the triangles, and tends to avoid sliver triangles.
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example,
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex.
In the mathematical disciplines of topology and geometry, an orbifold is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring of a triangulation of an -dimensional simplex contains a cell whose vertices all have different colors.
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon. Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points.
A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system of line segments with the points as endpoints, minimizing the total length of the segments. In it, any two points can reach each other along a path through the line segments. It can be found as the minimum spanning tree of a complete graph with the points as vertices and the Euclidean distances between points as edge weights.
A triangulation of a set of points in the Euclidean space is a simplicial complex that covers the convex hull of , and whose vertices belong to . In the plane, triangulations are made up of triangles, together with their edges and vertices. Some authors require that all the points of are vertices of its triangulations. In this case, a triangulation of a set of points in the plane can alternatively be defined as a maximal set of non-crossing edges between points of . In the plane, triangulations are special cases of planar straight-line graphs.
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or in modeling.
The art gallery problem or museum problem is a well-studied visibility problem in computational geometry. It originates from the following real-world problem:
"In an art gallery, what is the minimum number of guards who together can observe the whole gallery?"
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.
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. A typical 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.
In Euclidean plane geometry, a pseudotriangle (pseudo-triangle) is the simply connected subset of the plane that lies between any three mutually tangent convex sets. A pseudotriangulation (pseudo-triangulations) is a partition of a region of the plane into pseudotriangles, and a pointed pseudotriangulation is a pseudotriangulation in which at each vertex the incident edges span an angle of less than π.
In computational geometry, the Bowyer–Watson algorithm is a method for computing the Delaunay triangulation of a finite set of points in any number of dimensions. The algorithm can be also used to obtain a Voronoi diagram of the points, which is the dual graph of the Delaunay triangulation.
In mesh generation, Delaunay refinements are algorithms for mesh generation based on the principle of adding Steiner points to the geometry of an input to be meshed, in a way that causes the Delaunay triangulation or constrained Delaunay triangulation of the augmented input to meet the quality requirements of the meshing application. Delaunay refinement methods include methods by Chew and by Ruppert.
In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form
In computational geometry, a constrained Delaunay triangulation is a generalization of the Delaunay triangulation that forces certain required segments into the triangulation as edges, unlike the Delaunay triangulation itself which is based purely on the position of a given set of vertices without regard to how they should be connected by edges. It can be computed efficiently and has applications in geographic information systems and in mesh generation.
In mathematics, a flip graph is a graph whose vertices are combinatorial or geometric objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are special cases of geometric graphs.
In topological data analysis, the Vietoris–Rips filtration is the collection of nested Vietoris–Rips complexes on a metric space created by taking the sequence of Vietoris–Rips complexes over an increasing scale parameter. Often, the Vietoris–Rips filtration is used to create a discrete, simplicial model on point cloud data embedded in an ambient metric space. The Vietoris–Rips filtration is a multiscale extension of the Vietoris–Rips complex that enables researchers to detect and track the persistence of topological features, over a range of parameters, by way of computing the persistent homology of the entire filtration. It is named after Leopold Vietoris and Eliyahu Rips.
