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diagonal (Definition)

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.


\begin{pspicture} % latex2html id marker 74 (-8,0)(0,3) \pspolygon(-5,0)(-3,0)(-... ...){$X$} \rput[l](-6,1.5){.} \rput[a](-3,3){.} \rput[r](-2,1.4){.} \end{pspicture}
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}(-8,0)(0,2) \pspolygon(-5,0)(-4,0.5)(-2,0)(-2,2)(-3,1)(-4,1.3)(-5,1.3)(-6,2)(-6,0.7) \psline[linecolor=red](-6,0.7)(-2,2) \end{pspicture}
  • If a polygon $ P$ has $ n$ (distinct) vertices, then it has $ \displaystyle{\frac{n(n-3)}{2}}$ diagonals.



"diagonal" is owned by CWoo. [ full author list (2) ]
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See Also: polygon, polyhedron

Also defines:  adjacent vertices
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Cross-references: relative interior, convex, endpoints, hexagon, edge, line segment, vertices, polyhedron, polygon
There are 163 references to this entry.

This is version 4 of diagonal, born on 2007-10-12, modified 2007-10-29.
Object id is 9990, canonical name is Diagonal.
Accessed 1982 times total.

Classification:
AMS MSC51N05 (Geometry :: Analytic and descriptive geometry :: Descriptive geometry)

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