Vertex (geometry)

Source: Wikipedia, the free encyclopedia.
For vertices in the geometry of curves, see Vertex (curve). For other uses of the word, see Vertex (disambiguation).

In geometry, a vertex (pl.: vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect. 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.[1][2][3]

Definition

Of an angle

A vertex of an angle is the endpoint where two lines or rays come together.

The vertex of an

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.[3][4]

Of a polytope

A vertex is a corner point of a

intersection of edges, faces or facets of the object.[4]

In a polygon, a vertex is called "

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".[5] 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

1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope,[6]
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.[7] However, a smooth curve approximation to a polygon will also have additional vertices, at the points where its curvature is minimal.[citation needed]

Of a plane tiling

A vertex of a plane tiling or

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

open line segment
between C and D is entirely inside the polygon. Vertex C is a mouth, because the open line segment between A and B is entirely outside the polygon.

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.[9]

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) According to the two ears theorem, every simple polygon has at least two ears.[10]

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.

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

cube
has 12 edges and 6 faces, the formula implies that it has eight vertices.

Vertices in computer graphics

Main article: Vertex (computer graphics)

In

vertex shader, part of the vertex pipeline
.

See also

References

  1. ^ Weisstein, Eric W. "Vertex". MathWorld.
  2. ^ "Vertices, Edges and Faces". www.mathsisfun.com. Retrieved 2020-08-16.
  3. ^ a b "What Are Vertices in Math?". Sciencing. Retrieved 2020-08-16.
  4. ^
    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 0-486-60090-4
    (vol. 3).
  5. ^ Jing, Lanru; Stephansson, Ove (2007). Fundamentals of Discrete Element Methods for Rock Engineering: Theory and Applications. Elsevier Science.
  6. ISBN 0-521-81496-0
    (Page 29)
  7. ISBN 978-3-7643-8620-7
    .
  8. ISBN 0-12-040602-0
    , Academic Press, 1989.
  9. ISBN 978-0-691-14553-2
    .
  10. MR 0367792
    .
  11. ^ Christen, Martin. "Clockworkcoders Tutorials: Vertex Attributes". Khronos Group. Archived from the original on 12 April 2019. Retrieved 26 January 2009.

External links

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