For other uses of the word, see Vertex.

In geometry, a vertex (plural "vertices") is a special kind of point. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume

## Definitions

### Of an angle

The vertex of an angle is 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 and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called

### Of a polytope

A vertex is a corner point of a polygon, polyhedron, or other higher dimensional polytope, formed by the intersection of edges, faces or facets of the object: a vertex of a polygon is the point of intersection of two edges, a vertex of a polyhedron is the point of intersection of three or more edges or faces, and a vertex of a d-dimensional polytope is the intersection point of d or more edges, faces or factes. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations In Geometry, a face of a Polyhedron is any of the Polygons that make up its boundaries

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 radians; otherwise, it is called "concave" or "reflex". In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the 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 In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called 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. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope, and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices. For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out This article is not about the Topological skeleton concept of Computer graphics In Mathematics, particularly in Algebraic topology 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. In the geometry of Curves a vertex is a point of where the first derivative of Curvature is zero However, a smooth curve approximation to a polygon will also have additional vertices, at the points where its curvature is minimal.

### Of a plane tiling

A vertex of a plane tiling or 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. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps 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. In Topology, a CW complex is a type of Topological space introduced by J In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments

## Principal vertex

A polygon vertex xi 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 xi of a simple polygon P is called an ear if the diagonal [x(i − 1),x(i + 1)] that bridges xi lies entirely in P. (see also convex polygon)

### Mouths

A principal vertex xi of a simple polygon P is called a mouth if the diagonal [x(i − 1),x(i + 1)] lies 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. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions In the Geometry of Computer graphics, a vertex normal at a vertex of a Polyhedron is the normalized average of the Surface normals Vertex shader (abbreviation VS) is a shader program normally executed on the Graphics processing unit. 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