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.
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. 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, 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.
Of a plane tiling A vertex of a plane tiling or
tessellation is a point where three or more tiles meet; 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==