# Vertex (geometry) Information

*https://en.wikipedia.org/wiki/Vertex_(geometry)*

In
geometry, a **vertex** (in plural form: **vertices** or **vertexes**), often denoted by letters such as , , , ,^{
[1]} is a
point where two or more
curves,
lines, or
edges meet. As a consequence of this definition, the point where two lines meet to form an
angle and the corners of
polygons and
polyhedra are vertices.^{
[2]}^{
[3]}^{
[4]}

## Definition

### 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 intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place.^{
[5]}^{
[4]}

### 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.^{
[5]}

In a polygon, a vertex is called "
convex" if the
internal angle of the polygon (i.e., the
angle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two
right angles); otherwise, it is called "concave" or "reflex".^{
[6]} 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 is concave otherwise.

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,^{
[7]} 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.^{
[8]} 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;^{
[9]} 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.

## 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*.^{
[10]}

### 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) According to the
two ears theorem, every simple polygon has at least two ears.^{
[11]}

### Mouths

A principal vertex *x*_{i} of a simple polygon P is called a mouth if the diagonal [*x*_{(i − 1)}, *x*_{(i + 1)}] lies outside the boundary of P.

## Number of vertices of a polyhedron

Any convex polyhedron's surface has Euler characteristic

where *V* is the number of vertices, *E* is the number of
edges, and *F* is the number of
faces. This equation is known as
Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, since a
cube has 12 edges and 6 faces, the formula implies that it has 8 vertices.

## Vertices in computer graphics

In
computer graphics, objects are often represented as triangulated
polyhedra in which the
object 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 normal;^{
[12]} these properties are used in rendering by a
vertex shader, part of the
vertex pipeline.

## See also

## References

**^**"Compendium of Mathematical Symbols".*Math Vault*. 2020-03-01. Retrieved 2020-08-16.**^**Weisstein, Eric W. "Vertex".*MathWorld*.**^**"Vertices, Edges and Faces".*www.mathsisfun.com*. Retrieved 2020-08-16.- ^
^{a}^{b}"What Are Vertices in Math?".*Sciencing*. Retrieved 2020-08-16. - ^
^{a}^{b}Heath, Thomas L. (1956).*The Thirteen Books of Euclid's Elements*(2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.- (3 vols.): ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3).

**^**Jing, Lanru; Stephansson, Ove (2007).*Fundamentals of Discrete Element Methods for Rock Engineering: Theory and Applications*. Elsevier Science.**^**Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 29)**^**Bobenko, Alexander I.; Schröder, Peter; Sullivan, John M.; Ziegler, Günter M. (2008).*Discrete differential geometry*. Birkhäuser Verlag AG. ISBN 978-3-7643-8620-7.**^**M.V. Jaric, ed, Introduction to the Mathematics of Quasicrystals (Aperiodicity and Order, Vol 2) ISBN 0-12-040602-0, Academic Press, 1989.**^**Devadoss, Satyan; O'Rourke, Joseph (2011).*Discrete and Computational Geometry*. Princeton University Press. ISBN 978-0-691-14553-2.**^**Meisters, G. H. (1975), "Polygons have ears",*The American Mathematical Monthly*,**82**: 648–651, doi: 10.2307/2319703, MR 0367792.**^**Christen, Martin. "Clockworkcoders Tutorials: Vertex Attributes". Khronos Group. Retrieved 26 January 2009.