In discrete geometry, an arrangement is the decomposition of the d-dimensional linear, affine, or projective space into connected cells of different dimensions, induced by a finite collection of geometric objects, which are usually of dimension one less than the dimension of the space, and often of the same type as each other, such as hyperplanes or spheres.
For a set of objects in , the cells in the arrangement are the connected components of sets of the form for subsets of . That is, for each the cells are the connected components of the points that belong to every object in and do not belong to any other object. For instance the cells of an arrangement of lines in the Euclidean plane are of three types:
Of particular interest are the arrangements of lines and arrangements of hyperplanes.
More generally, geometers have studied arrangements of other types of curves in the plane, and of other more complicated types of surface. [1] Arrangements in complex vector spaces have also been studied; since complex lines do not partition the complex plane into multiple connected components, the combinatorics of vertices, edges, and cells does not apply to these types of space, but it is still of interest to study their symmetries and topological properties. [2]
An interest in the study of arrangements was driven by advances in computational geometry, where the arrangements were unifying structures for many problems. Advances in study of more complicated objects, such as algebraic surfaces, contributed to "real-world" applications, such as motion planning and computer vision. [3]
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, with any two points, it contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersect every line into a single line segment . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.
In geometry, the convex hull or 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 geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined.
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seems to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.
A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.
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.
Solid modeling is a consistent set of principles for mathematical and computer modeling of three-dimensional solids. Solid modeling is distinguished from related areas of geometric modeling and computer graphics by its emphasis on physical fidelity. Together, the principles of geometric and solid modeling form the foundation of 3D-computer-aided design and in general support the creation, exchange, visualization, animation, interrogation, and annotation of digital models of physical objects.
In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide all of them in half with a single (n − 1)-dimensional hyperplane.
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space. Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(Cn+1), Pn(C) or CPn. When n = 1, the complex projective space CP1 is the Riemann sphere, and when n = 2, CP2 is the complex projective plane.
In geometry an arrangement of lines is the partition of the plane formed by a collection of lines. Bounds on the complexity of arrangements have been studied in discrete geometry, and computational geometers have found algorithms for the efficient construction of arrangements.
In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S. Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space. One may ask how these properties are related to the arrangement and its intersection semilattice. The intersection semilattice of A, written L(A), is the set of all subspaces that are obtained by intersecting some of the hyperplanes; among these subspaces are S itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc.. These intersection subspaces of A are also called the flats ofA. The intersection semilattice L(A) is partially ordered by reverse inclusion.
The Szemerédi–Trotter theorem is a mathematical result in the field of combinatorial geometry. It asserts that given n points and m lines in the Euclidean plane, the number of incidences is
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 geometry, space partitioning is the process of dividing a space into two or more disjoint subsets. In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions.
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn. Some authors use the terms "convex polytope" and "convex polyhedron" interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope.
In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension d ≥ 3. Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid are:
In geometry, a set K ⊂ Rd is defined to be orthogonally convex if, for every line L that is parallel to one of standard basis vectors, the intersection of K with L is empty, a point, or a single segment. The term "orthogonal" refers to corresponding Cartesian basis and coordinates in Euclidean space, where different basis vectors are perpendicular, as well as corresponding lines. Unlike ordinary convex sets, an orthogonally convex set is not necessarily connected.
In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself a Euclidean space. The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes.
In mathematics, the vertex enumeration problem for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the vertices of a convex polytope specified by a set of linear inequalities:
In geometry, a Sylvester–Gallai configuration consists of a finite subset of the points of a projective space with the property that the line through any two of the points in the subset also passes through at least one other point of the subset.
