Vertex — corner point in geometry, polyhedra and graph theory
A vertex is a point where two or more edges, lines or curves meet. Used in polygons, polyhedra, graphs and computer graphics; described by angle, valence (degree) and local geometry.
A vertex (plural: vertices) is a fundamental geometric concept that denotes a corner or meeting point of two or more edges, lines or curves. In elementary geometry a vertex is the point at which the sides of a polygon or the edges of a polyhedron intersect. Vertices are usually labeled with capital letters such as P, Q, R and S; these labels are commonly shown near the corner points in diagrams.
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Definitions and basic properties
Formally, a vertex is any point at which two or more geometric objects meet. In the plane this typically means the intersection of two segments or rays; in three dimensions it means the intersection of three or more faces or edges. A polygon with n sides has n vertices. For a convex polyhedron the relationship between the number of vertices (V), edges (E) and faces (F) often satisfies the Euler characteristic V − E + F = 2, a basic topological identity for simple polyhedra.
Types of vertices and measures
Vertices can be classified by local geometry. In a polygon, a convex vertex has an interior angle less than 180°, while a concave vertex has an interior angle greater than 180°. The interior angles of a simple n-gon sum to (n − 2) × 180°. For polyhedra, the shape of the region around a vertex can be described by its vertex figure, the polygonal cross-section obtained by slicing close to the vertex and looking at how faces meet.
Vertex in graph theory and combinatorics
Outside classical geometry, the word "vertex" is used in graph theory to mean a node in a graph. There, a vertex is an abstract point connected to other vertices by edges; its degree (also called valence) is the number of incident edges. Graph vertices need not have coordinates; they represent relational structure in networks, circuits, maps and many algorithms.
Examples and common occurrences
- Triangle: 3 vertices.
- Square: 4 vertices.
- Regular pentagon: 5 vertices; each interior angle equals 540°/5.
- Tetrahedron: 4 vertices; a simple three-dimensional example of a polyhedron.
These concrete cases help bridge polygonal geometry and solid geometry: a cube has 8 vertices, a common fact in elementary solid geometry.
Applications and notable facts
Vertices appear in many domains. In computer graphics, vertices are the basic units used to build meshes for 3D models; vertex coordinates, normals and texture data are passed to rendering pipelines. In computational geometry, vertex enumeration and manipulation underpin algorithms for convex hulls, triangulations and mesh processing. In architecture and engineering, vertices identify structural joints and connection points.
When studying geometric objects it is useful to note whether vertices are simple (meeting of a small fixed number of edges), regular (symmetrically equivalent under the object’s symmetries) or singular (points where normal assumptions break down). Many standard texts and resources discuss these aspects in more detail; for basic definitions see entries on curves, rays and sides, and for polyhedra examples consult sources about the tetrahedron and the regular pentagon.
Overall, the vertex is a simple but versatile concept: geometrically it marks corners, combinatorially it marks nodes, and computationally it marks the primitives that underlie modelling and analysis.
Vertices in Linear Optimization
Corners play an important role in linear optimization, since it can be shown that the optimal function value is always assumed to be in a corner of the restriction set. The simplex algorithm in particular takes advantage of this by systematically running from corner to corner until it finds the optimal function value. The admissible base solutions used here are exactly the corners of the polyhedron.
Differentiation of corners
To distinguish between corners that are usually right-angled, we speak of "inside" and "outside" corners. In a convex polygon, the angles of the corners are always smaller than 180° when viewed from the inside and always larger than 180° when viewed from the outside. A corner is called an "inside corner" if its angle is less than 180°. Otherwise it is an "outside corner". Thus, in rooms, corners that you look into are inside corners and corners that protrude are outside corners. The observation is relative, that is, in relation to the object. The floor of a room lies with its outer corners in the inner corners of the room. These inner corners lie correspondingly at the outer corners of the floor.
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AlegsaOnline.com Vertex — corner point in geometry, polyhedra and graph theory Leandro Alegsa
URL: https://en.alegsaonline.com/art/104754