Ingeometry, avertex (pl.:vertices orvertexes), also called acorner, is apoint where two or morecurves,lines, orline segmentsmeet orintersect. For example, the point where two lines meet to form anangle and the point whereedges ofpolygons andpolyhedra meet are vertices.[1][2][3]
Thevertex of anangle is the point where tworays 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]
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A vertex is a corner point of apolygon,polyhedron, or other higher-dimensionalpolytope, formed by theintersection ofedges,faces or facets of the object.[4]
In a polygon, a vertex is called "convex" if theinternal angle of the polygon (i.e., theangle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, tworight 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 smallsphere centered at the vertex is convex, and is concave otherwise.
Polytope vertices are related tovertices of graphs, in that the1-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, ingraph 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 thevertices 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]
A vertex of a plane tiling ortessellation is a point where three or more tiles meet;[8] 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 topologicalcell complex, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such assimplicial complexes are its zero-dimensional faces.
A polygon vertexxi of asimple polygonP is a principal polygon vertex if thediagonal[x(i − 1),x(i + 1)] intersects the boundary ofP only atx(i − 1) andx(i + 1). There are two types of principal vertices:ears andmouths.[9]
Anyconvex polyhedron's surface hasEuler characteristic
whereV is the number of vertices,E is the number ofedges, andF is the number offaces. This equation is known asEuler'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 acube has 12 edges and 6 faces, the formula implies that it has eight vertices.
Incomputer graphics, objects are often represented as triangulatedpolyhedra in which theobject 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, andsurface normal.[11] These properties are used in rendering by avertex shader, part of thevertex pipeline.
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