In geometry, a vertex (plural "vertices") is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. For other uses, see Geometry (disambiguation). ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Look up polygon in Wiktionary, the free dictionary. ... For the game magazine, see Polyhedron (magazine). ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...

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 geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ... This article is about angles in geometry. ...

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. Look up Convex set in Wiktionary, the free dictionary. ... External angles law 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. ... This article is about angles in geometry. ... For other uses, see Sphere (disambiguation). ...

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. A tessellated plane seen in street pavement. ... In topology, a CW complex is a type of topological space introduced by J.H.C. Whitehead to meet the needs of homotopy theory. ... In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts. ...

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. This article just presents the basic definitions. ... In mathematics, particularly in algebraic topology, the 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. ... A ellipse (red) and its evolute (blue), the dots are the vertices of the curve, each vertex corresponds to a cusp on the evolute. ...

Principal vertex

A polygon vertex x_i of a simple polygon P is a principal polygon vertex if the diagonal [x_{(i − 1)},x_{(i + 1)}] intersects the boundary of P only at x_{(i − 1)} and x_{(i + 1)}. There are two types of principal vertices, ears and mouths.

Ears

A principal vertex x_i of a simple polygon P is called an ear if the diagonal [x_{(i − 1)},x_{(i + 1)}] that bridges x_i lies entirely in P. (see also convex polygon) A convex pentagon In geometry, a convex polygon is a simple polygon whose interior is a convex set. ...

Mouths

A principal vertex x_i of a simple polygon P is called a mouth if the diagonal [x_{(i − 1)},x_{(i + 1)}] if the interior of [x_{(i − 1)},x_{(i + 1)}] lies in the outside the boundary of P). (see also concave polygon) In geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. ...

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. This article is about the scientific discipline of computer graphics. ... 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. ... Vertex and pixel (or fragment) shaders are shaders that run on a graphics card, executed once for every vertex or pixel in a specified 3D mesh. ... 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...

External links

• Eric W. Weisstein, Polygon Vertex at MathWorld.
• Eric W. Weisstein, Polyhedron Vertex at MathWorld.
• Eric W. Weisstein, Principal Vertex at MathWorld.