vertex (geometry)

Information about vertex (geometry)

For other uses of the word, see Vertex.
In geometry, a vertex (plural "vertices") is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope.

Definition

For each of those figures, a vertex is a point formed by the intersection of faces of the object: a vertex of a polygon is the point of intersection of two polygon edges, a vertex of a polyhedron is the point of intersection of three or more polyhedron facets, and a vertex of a d-dimensional polytope is the intersection point of d or more polytope facets. A vertex can also refer to an angle, the point where two rays begin or meet, where two line segments join or meet, where two lines cross (intersect), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place.

In a polygon, a vertex is called "convex" if the internal angle of the polygon, that is, the angle formed by the two edges at the vertex, with the polygon inside the angle, is less than π; otherwise, it is called "concave" or "reflex". More generally, a vertex of a polyhedron or polytope is convex if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, and concave otherwise.

A vertex of a plane tessellation is a point where three or more tiles meet; generally, but not always, the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles. More generally, a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional faces.

Geometric vertices are related to vertices of graphs, in that the 1-skeleton of a polyhedron or polytope is a graph, the vertices of which correspond to the vertices of the polyhedron or polytope, and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices. However, in graph theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. There is also a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, and if a polygon is approximated by a smooth curve there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will also have additional vertices, at the points where its curvature is minimal.

Principal vertex

A polygon vertex of a simple polygon P is a principal polygon vertex if the diagonal intersects the boundary of P only at and . There are two types of principal vertices, ears and mouths.

Ears

A principal vertex of a simple polygon P is called an ear if the diagonal that bridges lies entirely in P. (see also convex polygon)

Mouths

A principal vertex of a simple polygon P is called a mouth if the diagonal if the interior of lies in the outside the boundary of P). (see also concave polygon)

Vertices in computer graphics

In computer graphics, objects are often represented as triangulated polyhedra in which the vertices are associated not only with three spatial coordinates but also with other graphical information necessary to render the object correctly, such as colors, reflectance properties, textures, and surface normals; these properties are used in rendering by a vertex shader, part of the vertex pipeline.

External links

Vertex (Latin: corner; plural vertices or vertexes) may refer to:

Mathematics

  • Vertex (geometry), a corner point of a polygon, polyhedron or general polytope, or by extension of a computer graphics point object.

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Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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A spatial point is a concept used to define an exact location in space. It has no volume, area or length, making it a zero dimensional object. Points are used in the basic language of geometry, physics, vector graphics (both 2D and 3D), and many other fields.
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POLYGONE is an Electronic Warfare Tactics Range located on the border between France and Germany. It is one of only two in Europe, the other being RAF Spadeadam.

The range, also referred to as the Multi-national Aircrew Electronic Warfare Tactics Facility (MAEWTF), is
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polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges.

The word polyhedron comes from the Classical Greek πολυεδρον, from poly-
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In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions. Beyond that, the term is used for a variety of related mathematical concepts.
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face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube. The suffix -hedron is derived from the Greek word hedra which means face.
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angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept
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convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex.
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In geometry, an interior angle (or internal angle) is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon. A simple polygon has exactly one internal angle per vertex.
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angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept
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A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface. In mathematics, a sphere is the set of all points in three-dimensional space (R3
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tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible.
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In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. The idea was to have a class of spaces that was broader than simplicial complexes (in modern language, which had better categorical properties), but still
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simplicial complex is a topological space of a particular kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts (see illustration).
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vertex (plural vertices) or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered
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n-skeleton of a topological space X presented as a simplicial complex, or CW complex, refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions m ≤ n.
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In the geometry of curves a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature. Other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant.
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In geometry, a convex polygon is a simple polygon whose interior is a convex set. The following properties of a simple polygon are all equivalent to convexity:
  • Every internal angle is at most 180 degrees.

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In geometry, a convex polygon is a simple polygon whose interior is a convex set. The following properties of a simple polygon are all equivalent to convexity:
  • Every internal angle is at most 180 degrees.

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Computer graphics is a sub-field of computer science and is concerned with digitally synthesizing and manipulating visual content. Although the term often refers to three-dimensional computer graphics, it also encompasses two-dimensional graphics and image processing.
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In the geometry of computer graphics, a vertex normal at a vertex of a polyhedron is the normalized average of the surface normals of the faces that contain that vertex. The average can be weighted by the area of the face or it can be unweighted.
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Vertex shader (abbreviation VS) is a shader program, normally executed on the Graphics processing unit.

Function

A vertex shader is a graphics processing function used to add special effects to objects in a 3D environment by performing mathematical operations on
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The function of the vertex pipeline in any GPU is to take geometry data (usually supplied as vector points), work with it if needed with either fixed function processes (earlier DirectX), or a vertex shader program (later DirectX), and create all of the 3D data points in a scene to
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Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He currently works for Wolfram Research, Inc.
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MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc., the creators of the Mathematica computer algebra system. It is also partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at
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Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He currently works for Wolfram Research, Inc.
..... Click the link for more information.
MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc., the creators of the Mathematica computer algebra system. It is also partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at
..... Click the link for more information.
Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He currently works for Wolfram Research, Inc.
..... Click the link for more information.
MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc., the creators of the Mathematica computer algebra system. It is also partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at
..... Click the link for more information.

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