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| Title of object: |
polygon |
| Canonical Name: |
Polygon |
| Type: |
Definition |
| Created on: |
2002-01-05 22:14:01 |
| Modified on: |
2007-06-30 01:02:34 |
| Classification: |
msc:51-00, msc:51G05 |
| Defines: |
side, edge, vertex, vertices, angle sum, polygonal curve, perimeter |
Preamble:
\usepackage{graphicx}
%\usepackage{xypic}
\usepackage{amsthm}
\usepackage{bbm}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\mathbb}[1]{\mathbbmss{#1}}
\newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}}
\newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}}
%\theoremstyle{definition}
%\newtheorem{Definition}{Definition} |
Content:
\section{Definitions}
A \emph{polygonal curve} is a \PMlinkname{simple closed path}{Curve}
consisting of a finite sequence of coplanar points together with each
\PMlinkname{open interval}{OrderedGeometry} determined by two
consecutive points on the path. A \emph{polygon} is a closed
\PMlinkname{planar}{IncidenceGeometry} region bounded by a polygonal
curve. Each closed interval in the polygonal curve is called an
\emph{edge} or \emph{side} of the polygon, and each point in the
sequence of points determining the polygonal curve is called a
\emph{vertex} of the polygon. Each polygon $P$ can be dissected into a
set of triangles, $T$ such that the vertices of the triangles in $T$
are vertices of $P$. The sum of the measures of the angles of the triangles in $T$
is called the \emph{angle sum} of $P$.
\figuraex{polygons}{scale=0.75}
A polygon with $n$ sides is called an $n$-gon, although for small $n$
there are more traditional names:
\begin{center}
\begin{tabular}{||c|c||} \hline
Number of sides& Name of the polygon \\ \hline
3 & triangle \\
4 & quadrilateral \\
5 & pentagon\\
6 & hexagon \\
7 & heptagon\\
8 & octagon\\
9 & nonagon\\
10 & decagon\\
11 & hendecagon, undecagon\\
12 & dodecagon\\\hline
\end{tabular}
\end{center}
Below are some properties for polygons.
\begin{enumerate}
\item In a Euclidean space, the angle sum of an
$n$-gon is $(n-2)\pi$.
\item In Euclidean geometry the boundary of a polygon divides the plane into two connected
components, one bounded (the interior of the polygon) and one
unbounded. This result is the Jordan curve theorem for polygons.
Moise proves this directly in \cite{moise}, pp. 16-18.
%A direct proof can be
%found in \cite{moise}, pp. 16--18.
\item In complex analysis, the Schwarz-Christoffel transformation
\cite{silverman} gives a conformal map from any polygon to the upper
half plane.
\item The area of a lattice polygon can be calculated using Pick's
theorem.
\end{enumerate}
The \emph{perimeter} of a polygon is the sum of the lengths of the
line segments in its bounding polygonal curve.
Some authors do not include the interior of a polygon as part of the
polygon, and thus identify a polygon with what we call a polygonal
curve. Such authors sometimes remove the requirement that the path
determining a polygon be simple.
\begin{thebibliography}{9}
\bibitem{borsuk-szmielew}
K. Borsuk and W. Szmielew, \emph{Foundations of Geometry},
North-Holland Publishing Company, 1960.
\bibitem{forder}
H.G. Forder, \emph{The Foundations of Euclidean Geometry},
Dover Publications, 1958.
\bibitem{moise}
E.E. Moise, \emph{Geometric Topology in Dimensions 2 and 3},
Springer-Verlag, 1977.
\bibitem{silverman}
R.A. Silverman, \emph{Introductory Complex Analysis},
Dover Publications, 1972.
\end{thebibliography}
\PMlinkescapeword{segments}
\PMlinkescapeword{maximal}
\PMlinkescapeword{name}
\PMlinkescapeword{names}
\PMlinkescapeword{meet}
\PMlinkescapeword{opens}
\PMlinkescapeword{properties}
\PMlinkescapeword{divides}
\PMlinkescapeword{bounded}
\PMlinkescapeword{unbounded}
\PMlinkescapeword{complex}
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