Vertex

A vertex is a point of significance in various branches of mathematics and related fields. The term is used to describe a point where geometric elements meet, such as the corners of polygons, as well as the fundamental unit of a graph in graph theory, and a point in 3D space with attributes in computer graphics.

In geometry, a vertex of a polygon is a corner where two edges meet. In a polyhedron, a vertex is a point where at least three faces meet. The degree of a vertex, when defined for geometric graphs, is the number of incident edges or, in some contexts, the number of neighboring vertices.

In graph theory, a vertex (also called a node) is one of the primary components of a graph. It is connected to other vertices by edges. Vertices may carry data such as labels, weights, or coordinates. In directed graphs, vertices have in-degrees and out-degrees, reflecting the number of incoming and outgoing edges. Graphs can be used to model networks, relationships, and many discrete structures.

In computer graphics, a vertex represents a point in 3D space and typically stores additional attributes such as color, a normal vector, and texture coordinates. A set of vertices forms a vertex buffer, and vertices undergo processing in the graphics pipeline, including transformations and shading, often via a vertex shader before rasterization.

In the context of conic sections, the vertex is a point where curvature is extremal. For a parabola, the vertex is the point where the axis of symmetry intersects the curve, and it appears in the standard vertex form y = a(x − h)^2 + k, which places the vertex at (h, k).