diagonal

Let $P$ be a polygon or a polyhedron. Two vertices on $P$ are adjacent if the line segment joining them is an edge of $P$. A diagonal of $P$ is a line segment joining two non-adjacent vertices.

Below is a figure showing a hexagon and all its diagonals (in red) with $X$ as one of its endpoints.

Remarks.

• If $P$ is convex, then the relative interior of a diagonal lies in the relative interior of $P$. Below is a figure showing that a diagonal may partially lie outside of $P$.

  $$\begin{pspicture}(-227.62204pt,0.0pt)(0.0pt,56.905502pt)\leavevmode% \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \special{pst: \pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end }\ignorespaces\leavevmode\ignorespaces\ignorespaces\ignorespaces\ignorespaces% \special{pst: \pst@dict\tx@STP\pst@newpath\psk@origin\psk@swapaxes\pst@code end }\ignorespaces\end{pspicture}$$

• If a polygon $P$ has $n$ (distinct) vertices, then it has ${\displaystyle \frac{n(n-3)}{2}}$ diagonals.

Title	diagonal
Canonical name	Diagonal
Date of creation	2013-03-22 17:34:41
Last modified on	2013-03-22 17:34:41
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 51N05
Related topic	BasicPolygon
Related topic	Polyhedron
Defines	adjacent vertices