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<br>Vertex – a point at which two curves, two straight lines, or two edges converge. It follows from this definition that the point at which two rays converge, forming an angle, is a vertex, and it is also the corner points of polygons and polyhedra.<br>DefinitionTop corner<br>The vertex of the corner is the point where the two rays originate; where two line segments converge; where two lines intersect; where is any combination of rays, segments and straight lines that form two (rectilinear) „sides” that converge at one point.<br>Vertex of a polygon of a polyhedron<br>A vertex is a corner point of a polygon or polyhedron (of any dimension), in other words, its 0-dimensional face.<br><br>In a polygon, a vertex is called „convex” if the inner angle of the polygon is less than π radians (180 ° is two right angles). Otherwise, the vertex is called “concave”.<br><br>More generally, the vertex of a polyhedron is convex if the intersection of the polyhedron with a sufficiently small sphere having the vertex as the center is a convex figure; otherwise, the vertex is concave.<br><br>The vertices of a polytope are associated with the vertices of the graph, since a polytope is a graph whose vertices correspond to the vertices of a polyhedron, and therefore, the graph of a polyhedron can be considered as a one-dimensional simplicial complex, the vertices of which are the vertices of the graph. However, in graph theory, vertices can have less 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 extremum points of its curvature – the vertices of the polygon are, in a sense, points of infinite curvature, and if the polygon is by a smooth curve, the points of extreme curvature will lie near the vertices of the polygon. However, approximating a polygon with a smooth curve gives additional vertices at points of minimum curvature.<br>Vertices of flat mosaics<br>The vertex of a flat mosaic (tiling) is the point where three or more mosaic tiles meet, but not only: the tiles of the paving are also polygons, and the vertices of the mosaic are the vertices of these tiles. More generally, tiling can be viewed as a kind of topological CW complex. of other types of complexes, such as simplicial ones, are faces of dimension zero.<br>Main peak<br>A vertex xi displaystyle x_ i of a P displaystyle P is a major vertex if the diagonal [ x i − 1 , x i + 1 ] [x_i-1,x_i+1] crosses the boundaries of P displaystyle P only at points xi – 1 displaystyle x_ i-1 and xi + 1 displaystyle x_ i + 1. There are two types of basic vertices: „ears” and „mouths” (see below).<br>„Ears”<br>The main vertex xi displaystyle x_ i of a simple polygon P displaystyle P is called an „ear” if the diagonal [ x i − 1 , x i + 1 ] displaystyle [x_i-1,x_i+1] lies entirely in P displaystyle P. (see also convex polygon)<br>„Mouths”<br>The main vertex xi displaystyle x_ i of a simple polygon P displaystyle P is called a „mouth” if the diagonal [ x i − 1 , x i + 1 ] displaystyle [x_i-1,x_i+1] lies outside P displaystyle P.<br>Number of vertices of a polyhedron<br>Any surface of a three-dimensional convex polyhedron has an Euler characteristic:<br><br> V – E + F = 2, displaystyle V-E + F = 2,<br><br>where V displaystyle V is the number of vertices, E displaystyle E is the number of edges, and F displaystyle F is the number of faces. This equality is known as the Euler equation. For example, a cube has 12 edges and 6 faces, so there are 8 vertices: 8 – 12 + 6 = 2 displaystyle 8-12 + 6 = 2.<br>Vertices in computer graphics<br>In computer graphics, objects are often represented as triangulated polyhedrons, in which not only three spatial coordinates are assigned to the vertices of an object, but also other graphic information necessary for the correct construction of an object image, such as color, reflectivity, texture, vertex normals. These properties are used when rendering with the vertex shader, part of the vertex processor.<br>
