Vertex (geometry)

In geometry, a vertex (pl.: vertices or vertexes), also called a corner, is a point where two or more curves, lines, or line segments meet or intersect. Vertices are important because they help us describe and understand shapes. For example, the point where two lines meet to form an angle is a vertex.

Vertices are also found in polygons and polyhedra. Polygons are flat, two-dimensional shapes like triangles and squares, and the points where their edges meet are vertices. In three-dimensional shapes, such as cubes or pyramids, the points where the edges come together are also called vertices.

These points are essential in many areas of math and science. They help us calculate sizes, angles, and areas, and they are used in computer graphics to create the shapes we see in videos and games. Understanding vertices gives us a foundation for exploring more complex geometric ideas. For vertices in the geometry of curves, see Vertex (curve). For other uses of the word, see Vertex (disambiguation).

Definition

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.

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. In a polygon, a vertex is called "convex" if the internal angle of the polygon is less than π radians (180°, two right angles); otherwise, it is called "concave" or "reflex".

Principal vertex

A polygon vertex x_i of a simple polygon is called a principal vertex if a line connecting the points next to it only touches the polygon at those two points. There are two special types of principal vertices: ears and mouths. An ear is when this connecting line lies completely inside the polygon. Thanks to a rule called the two ears theorem, every simple polygon has at least two ears. A mouth is when this connecting line lies outside the polygon's boundary.

Number of vertices of a polyhedron

A convex polyhedron follows a special rule called Euler's polyhedron formula: V − E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This means the number of vertices is connected to the edges and faces in a clear way. For example, a cube has 12 edges and 6 faces, so using the formula, we find it has eight vertices.

Vertices in computer graphics

Main article: Vertex (computer graphics)

In computer graphics, objects are shown as shapes made of triangles. The points, or object vertices, of these shapes have special details like colors and how they reflect light. These details help computers draw the objects correctly on a screen. A part called the vertex shader works with these points to make the images look realistic.