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Polygons
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polygon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon Types
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equiangular Polygon
In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal (that is, if it is also equilateral polygon, equilateral) then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths. For clarity, a planar equiangular polygon can be called ''direct'' or ''indirect''. A direct equiangular polygon has all angles turning in the same direction in a plane and can include multiple turn (angle), turns. Convex equiangular polygons are always direct. An indirect equiangular polygon can include angles turning right or left in any combination. A skew polygon, skew equiangular polygon may be isogonal figure, isogonal, but can't be considered direct since it is nonplanar. A spirolateral ''n''θ is a special case of an ''equiangular polygon'' with a set of ''n'' integer edge lengths repeating sequence until returning to the start, with vertex internal angles θ. Construc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew Polygon
In geometry, a skew polygon is a closed polygonal chain in Euclidean space. It is a figure (geometry), figure similar to a polygon except its Vertex (geometry), vertices are not all coplanarity, coplanar. While a polygon is ordinarily defined as a plane figure, the edge (geometry), edges and vertices of a skew polygon form a space curve. Skew polygons must have at least four vertices. The ''interior'' surface (geometry), surface and corresponding area measure of such a polygon is not uniquely defined. Skew infinite polygons (apeirogons) have vertices which are not all colinear. A zig-zag skew polygon or antiprismatic polygonRegular complex polytopes, p. 6 has vertices which alternate on two parallel planes, and thus must be even-sided. Regular skew polygons in 3 dimensions (and regular skew apeirogons in two dimensions) are always zig-zag. Skew polygons in three dimensions A regular skew polygon is a faithful symmetric realization of a polygon in dimension greater than 2. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Polytope
In geometry, a complex polytope is a generalization of a polytope in real coordinate space, real space to an analogous structure in a Complex number, complex Hilbert space, where each real dimension is accompanied by an imaginary number, imaginary one. A complex polytope may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Precise definitions exist only for the #Regular complex polytopes, regular complex polytopes, which are Configuration (polytope), configurations. The regular complex polytopes have been completely characterized, and can be described using a symbolic notation developed by Harold Scott MacDonald Coxeter, Coxeter. Some complex polytopes which are not fully regular have also been described. Definitions and introduction The complex line \mathbb^1 has one dimension with real number, real coordinates and another with imaginary number, imaginary coor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can arise through truncation operations on regular simple or star polygons. Branko Grünbaum identified two primary usages of this terminology by Johannes Kepler, one corresponding to the regular star polygons with List of self-intersecting polygons, intersecting edges that do not generate new vertices, and the other one to the isotoxal Concave polygon, concave simple polygons.Grünbaum & Shephard (1987). Tilings and Patterns. Section 2.5 Polygram (geometry), Polygrams include polygons like the pentagram, but also compound figures like the hexagram. One definition of a ''star polygon'', used in turtle graphics, is a polygon having ''q'' ≥ 2 Turn (geometry), turns (''q'' is called the turning number or Density (polygon), density), like in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex polygon, convex'' or ''star polygon, star''. In the limit (mathematics), limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a Line (geometry), straight line), if the edge length is fixed. General properties These properties apply to all regular polygons, whether convex or star polygon, star: *A regular ''n''-sided polygon has rotational symmetry of order ''n''. *All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. *Together with the property of equal-length sides, this implies that every regular polygon also h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can arise through truncation operations on regular simple or star polygons. Branko Grünbaum identified two primary usages of this terminology by Johannes Kepler, one corresponding to the regular star polygons with List of self-intersecting polygons, intersecting edges that do not generate new vertices, and the other one to the isotoxal Concave polygon, concave simple polygons.Grünbaum & Shephard (1987). Tilings and Patterns. Section 2.5 Polygram (geometry), Polygrams include polygons like the pentagram, but also compound figures like the hexagram. One definition of a ''star polygon'', used in turtle graphics, is a polygon having ''q'' ≥ 2 Turn (geometry), turns (''q'' is called the turning number or Density (polygon), density), like in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle (180 degrees or π radians). The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the ''base'', in which case the opposite vertex is called the ''apex''; the shortest segment between the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''Polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Polygon
In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons. The sum of external angles of a simple polygon is 2\pi. Every simple polygon with n sides can be polygon triangulation, triangulated by n-3 of its diagonals, and by the art gallery theorem its interior is visible from some \lfloor n/3\rfloor of its vertices. Simple polygons are commonly seen as the input to computational geometry problems, including point in polygon testing, area computation, the convex hull of a simple polygon, triangulation, and Euclidean shortest paths. Other constructions in geometry related to simple polygons include Schwarz–Christoffel mapping, used to find conformal maps involving simple polygons, polygonalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concave Polygon
A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180° degrees and 360° degrees exclusive. Polygon Some lines containing interior points of a concave polygon intersect its boundary at more than two points. Some diagonals of a concave polygon lie partly or wholly outside the polygon. Some sidelines of a concave polygon fail to divide the plane into two half-planes one of which entirely contains the polygon. None of these three statements holds for a convex polygon. As with any simple polygon, the sum of the internal angles of a concave polygon is (''n'' − 2) radians, equivalently 180°(''n'' − 2) degrees, where ''n'' is the number of sides. It is always possible to partition a concave polygon into a set of convex polygons. A polynomial-time algorithm for finding a decomposition into as few convex pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangential Polygon
In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices. All triangles are tangential, as are all regular polygons with any number of sides. A well-studied group of tangential polygons are the tangential quadrilaterals, which include the rhombi and kites. Characterizations A convex polygon has an incircle if and only if all of its internal angle bisectors are concurrent. This common point is the ''incenter'' (the center of the incircle). There exists a tangential polygon of ''n'' sequential sides ''a''1, ..., ''a''''n'' if and only if the system of equations :x_1+x_2=a_1,\quad x_2+x_3=a_2,\quad \ldots,\quad x_n+x_1=a_n has a solution (''x''1, ..., ''x''''n'') in pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
