Vertex (geometry)

In geometry, a vertex is the type of point at which two curves , two straight lines, or two edges converge. From this definition it follows that the point at which two rays converge, forming an angle , is a vertex, and also it is the corner points of polygons and polyhedra ^[1] .

Content

Definition

Vertex

The vertex of an angle is the point where two rays originate.

The top of the corner is the point where two rays come from; where the two segments meet; where two lines intersect; where any combination of rays, segments and straight lines forming two (rectilinear) "sides" that converge at one point ^[2] .

The top of the polygon polyhedron

A vertex is the corner point of a polygon or polyhedron (of any dimension), in other words, its 0-dimensional face .

In a polygon, a vertex is called “ convex ” if the internal angle of the polygon is less than π radians (180 ° - two right angles ). Otherwise, the vertex is called “concave”.

More generally, a vertex of a polyhedron is convex if the intersection of the polyhedron with a sufficiently small sphere having a vertex as the center is a convex figure; otherwise, the vertex is concave.

The vertices of the polyhedron are connected with the vertices of the graph , since the polyhedron is a graph whose vertices correspond to the vertices of the polyhedron ^[3] , and therefore, the polyhedron graph can be considered as a one-dimensional simplicial complex whose vertices are the vertices of the graph. However, in graph theory, vertices can have fewer than two incident edges , which is usually not allowed for geometric vertices. There is also a connection between the geometric vertices and the vertices of the curve , the points of the extrema of its curvature - the vertices of the polygon are, in a sense, points of infinite curvature, and if you approximate the polygon by a smooth curve, the points of extreme curvature will lie near the vertices of the polygon ^[4] . However, the approximation of a polygon using a smooth curve gives additional vertices at points of minimal curvature.

Flat Mosaic Tops

The top of a flat mosaic ( tiling ) is the point where three or more mosaic tiles meet ^[5] , but not only: the tiling tiles are also polygons, and the vertices of the mosaic are the vertices of these tiles. More generally, tiling can be considered as a type of topological CW complex . The vertices of other types of complexes, such as simplicial ones , are faces of zero dimension.

Main peak

The vertex B is an “ear” because the open segment between the vertices C and D lies completely inside the polygon. The vertex C is the “mouth”, since the open segment between A and B lies completely outside the polygon.

Vertex $x_{i}$ ${\ displaystyle x_ {i}}$ simple polygon $P$ ${\ displaystyle P}$ is the main vertex if the diagonal $[x_{i-1},x_{i+1}]$ ${\ displaystyle [x_ {i-1}, x_ {i + 1}]}$ crosses borders $P$ ${\ displaystyle P}$ only at points $x_{i-1}$ ${\ displaystyle x_ {i-1}}$ and $x_{i+1}$ ${\ displaystyle x_ {i + 1}}$ . There are two types of main vertices: “ears” and “mouths” (see below) ^[6] .

Ears

Main peak $x_{i}$ ${\ displaystyle x_ {i}}$ simple polygon $P$ ${\ displaystyle P}$ called “ear” if the diagonal $[x_{i-1},x_{i+1}]$ ${\ displaystyle [x_ {i-1}, x_ {i + 1}]}$ lies completely in $P$ ${\ displaystyle P}$ . (see also convex polygon )

Mouths

Main peak $x_{i}$ ${\ displaystyle x_ {i}}$ simple polygon $P$ ${\ displaystyle P}$ called “mouth” if the diagonal $[x_{i-1},x_{i+1}]$ ${\ displaystyle [x_ {i-1}, x_ {i + 1}]}$ lies out $P$ ${\ displaystyle P}$ .

The number of vertices of a polyhedron

Any surface of a three-dimensional convex polyhedron has an Euler characteristic :

V-E+F=2,

{\ displaystyle V-E + F = 2,}

Where $V$ ${\ displaystyle V}$ Is the number of vertices $E$ ${\ displaystyle E}$ Is the number of edges, and $F$ ${\ displaystyle F}$ - the number of faces. This equality is known as the Euler equation . For example, a cube has 12 edges and 6 faces, and therefore - 8 vertices: $8-12+6=2$ ${\ displaystyle 8-12 + 6 = 2}$ .

Vertices in computer graphics

In computer graphics, objects are often represented as triangulated polyhedra , in which not only three spatial coordinates are mapped to the vertices of the object , but also other graphic information necessary for the correct construction of the image of the object, such as color, reflectivity , texture , vertex normals ^[7] . These properties are used when constructing an image using the vertex shader , part of .

Notes

↑ Weisstein, Eric W. Vertex on the Wolfram MathWorld website.
↑ Heath, 1956 .
↑ McMullen, Schulte, 2002 , p. 29.
↑ Bobenko, Schröder, Sullivan, Ziegler, 2008 .
↑ Jaric, 1989 , p. 9.
↑ Devadoss, O'Rourke, 2011 .
↑ Christen, 2009 .

Literature

Thomas L. Heath. The Thirteen Books of Euclid's Elements. - 2nd ed. - New York: Dover Publications, 1956. - ISBN v1: 0-486-60088-2, v2: 0-486-60089-0, v3: 0-486-60090-4. (An authentic translation of the book of Euclid's "Beginnings" with extensive historical research and detailed comments on the text of the book.)
Lanru Jing, Ove Stephansson. Fundamentals of Discrete Element Methods for Rock Engineering: Theory and Applications. - 2007. - ISBN 978-0-444-82937-5 .
Peter McMullen, Egon Schulte. Abstract Regular Polytopes. - Cambridge University Press, 2002. - ISBN 0-521-81496-0 .
Introduction to the Mathematics of Quasicrystals / MV Jaric. - Academic Press, 1989. - T. 2. - (Aperiodicity and Order). - ISBN 0-12-040602-0 .
Alexander I. Bobenko, Peter Schröder, John M. Sullivan, Discrete differential geometry. - Birkhäuser Verlag AG, 2008 .-- ISBN 978-3-7643-8620-7 .
Satyan Devadoss, Joseph O'Rourke. Discrete and Computational Geometry . - Princeton University Press , 2011 .-- ISBN 978-0-691-14553-2 .
Martin Christen. Clockworkcoders Tutorials: Vertex Attributes. - Khronos Group , 2009.

Links

Weisstein, Eric W. Polygon Vertex on Wolfram MathWorld .
Weisstein, Eric W. Polyhedron Vertex on Wolfram MathWorld .
Weisstein, Eric W. Principal Vertex on Wolfram MathWorld .

[1] Weisstein, Eric W. Vertex on the Wolfram MathWorld website.

[_b80bcc26ab429c2a-2] Heath, 1956 .

[_4413a334c8aaacaf-3] McMullen, Schulte, 2002 , p. 29.

[_43dbc775f73bde97-4] Bobenko, Schröder, Sullivan, Ziegler, 2008 .

[_0a68f4a43f52dc92-5] Jaric, 1989 , p. 9.

[_347d8a7bc531f876-6] Devadoss, O'Rourke, 2011 .

[_8572a2539ceeb26c-7] Christen, 2009 .