# 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 Diagonal 2013-03-22 17:34:41 2013-03-22 17:34:41 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 51N05 BasicPolygon Polyhedron adjacent vertices