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Version 23 |
| \section{Definitions} |
\section{Definitions} |
| We follow Forder \cite{forder} for most of this entry. |
A \emph{polygonal curve} is a \PMlinkname{simple closed path}{Curve} |
| The term polygon can be defined if one has a definition of an interval. For this |
consisting of a finite sequence of coplanar points together with each |
| entry the geometry is called betweenness geometry. A betweenness geometry |
\PMlinkname{open interval}{OrderedGeometry} determined by two |
| is just one for which there is set of points and a betweenness relation $B$ defined. |
consecutive points on the path. A \emph{polygon} is a closed |
| Rather than write $(a,b,c) \in B$ we write a*b*c. |
\PMlinkname{planar}{IncidenceGeometry} region bounded by a polygonal |
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curve. Each closed interval in the polygonal curve is called an |
| \begin{enumerate} |
\emph{edge} or \emph{side} of the polygon, and each point in the |
| \item If $a$ and $b$ are distinct points the \emph{line $ab$} is the set of |
sequence of points determining the polygonal curve is called a |
| all points $p$ such that $p*a*b$ or $a*p*b$ or $a*b*p$. It can be shown |
\emph{vertex} of the polygon. Each polygon $P$ can be dissected into a |
| that the line $ab$ and the line $ba$ are the same set of points. |
set of triangles, $T$ such that the vertices of the triangles in $T$ |
| \item If $o$ and $a$ are distinct points A \emph{ray $oa$} is the set of all points $p$ such that |
are vertices of $P$. The sum of the measures of the angles of the triangles in $T$ |
| $p=o$ or $o*p*a$ or $o*a*p$. |
is called the \emph{angle sum} of $P$. |
| \item If $a$ and $b$ are distinct points, the \emph{open interval} is the set of points |
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| $p$ such that $a*p*b$. It is denoted by $(a,b).$ |
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| \item If $a$ and $b$ are distinct points, the \emph{closed interval} is |
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| $(a,b) \cup \{a\} \cup \{b\}$, and denoted by $[a,b].$ |
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| \item The \emph{way $a_1a_2\ldots a_n$}is the finite set of points $\{a_1, \ldots , a_n\}$ |
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| along with the open intervals $(a_1, a_2), (a_2,a_3), \ldots, (a_{n-1}, a_n)$. |
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| The points $a_1, \ldots, a_n$ are called the \emph{vertices} of the way, and the |
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| open intervals are called the \emph{sides} of the way. |
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| The closed intervals $[a_1,a_2], \ldots, [a_{n-1},a_n]$ are called the \emph{side-intervals} of |
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| the way. The lines $a_1a_2, \ldots , a_{n-1}a_n$ are called the \emph{side-lines} |
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| of the way. |
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| The way $a_1a_2\ldots a_n$ is said to \emph{join} $a_1$ to $a_n$. |
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| It is assumed that $a_{i-1}, a_i, a_{i+1}$ are not collinear. |
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| \item A way is said to be \emph{simple} if it does not meet itself. To be precise, |
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| (i) no two side-intervals meet in any point which is not a vertex, and (ii) no three side-intervals |
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| meet in any point. |
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| \item A \emph{polygon} is a way $a_1 a_2 \ldots a_n$ for which $a_1 = a_n$. Notice that there is |
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| no assumption that the points are coplanar. |
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| \item A \emph{simple polygon} is polygon for which the way is simple. |
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| \item A \emph{region} is a set of points not all collinear, any two of which can be joined by points of a way using |
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| only points of the region. |
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| \item A region $R$ is \emph{convex} if for each pair of points $a,b \in R$ the open interval $(a,b)$ is |
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| contained in $R.$ |
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| \item Let $X$ and $Y$ are two sets of points. If there is a set of points $S$ such that every way |
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| joining a point of $X$ to a point of $Y$ meets $S$ then $S$ is said to \emph{separate} |
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| $X$ from $Y$. |
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| \end{enumerate} |
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|
|
| Now assume that all points of the geometry are in one plane. Let $P$ be a polygon. ($P$ is called |
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| a plane polygon.) |
|
| \begin{enumerate} |
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| \item A ray or line which does not go through a vertex of $P$ will be called \emph{suitable}. |
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| \item An \emph{inside point} $a$ of $P$ is one for which a suitable ray from $a$ |
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| meets $P$ an odd number of times. Points that are not on or inside $P$ are said to be \emph{outside} |
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| $P$. |
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| \item Let $\{P_i\}$ be a set of polygons. We say that $\{P_i\}$ \emph{dissect} $P$ if the following |
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| three conditions are satisfied: (1) $P_i$ and $P_j$ do not have a common inside point for $i \not = j$, |
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| (ii) each inside point of $P$ is inside or on some $P_i$ and (iii) each inside point of $P_i$ is |
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| inside $P$. |
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| \item A \emph{convex polygon} is one whose inside points are all on the same side of any side-line |
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| of the polygon. |
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| \end{enumerate} |
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| \section{Theorems} |
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| Assume that all points are in one plane. Let $P$ be a polygon. |
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| \begin{enumerate} |
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| \item It can be shown that $P$ separates the other points of the plane into at least two regions and that |
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| if $P$ is simple there are exactly two regions. Moise proves this directly in \cite{moise}, pp. 16-18. |
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| \item It can be shown that $P$ can be dissected into triangles $\{T_i\}$ such that |
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| every vertex of a $T_i$ is a vertex of $P$. |
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| \item The following theorem of Euler can be shown: Suppose $P$ is dissected into $f>1$ polygons |
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| and that the total number of vertices of these polygons is $v$, and the number of open intervals |
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| which are sides is $e$. Then |
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| $$ |
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| v-e+f = 1 |
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| $$. |
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| \end{enumerate} |
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| |
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| \figuraex{polygons}{scale=0.75} |
\figuraex{polygons}{scale=0.75} |
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A plane polygon with $n$ sides is called an $n$-gon, although for small $n$
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A p polygon with $n$ sides is called an $n$-gon, although for small $n$
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| there are more traditional names: |
there are more traditional names: |
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|
| \begin{center} |
\begin{center} |
| \begin{tabular}{||c|c||} \hline |
\begin{tabular}{||c|c||} \hline |
| Number of sides& Name of the polygon \\ \hline |
Number of sides& Name of the polygon \\ \hline |
| 3 & triangle \\ |
3 & triangle \\ |
| 4 & quadrilateral \\ |
4 & quadrilateral \\ |
| 5 & pentagon\\ |
5 & pentagon\\ |
| 6 & hexagon \\ |
6 & hexagon \\ |
| 7 & heptagon\\ |
7 & heptagon\\ |
| 8 & octagon\\ |
8 & octagon\\ |
| 9 & nonagon\\ |
9 & nonagon\\ |
| 10 & decagon\\ |
10 & decagon\\ |
| 11 & hendecagon, undecagon\\ |
11 & hendecagon, undecagon\\ |
| 12 & dodecagon\\\hline |
12 & dodecagon\\\hline |
| \end{tabular} |
\end{tabular} |
| \end{center} |
\end{center} |
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Below are some properties for polygons. |
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\begin{enumerate} |
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\item In a Euclidean space, the angle sum of an |
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$n$-gon is $(n-2)\pi$. |
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\item In Euclidean geometry the boundary of a polygon divides the plane into two connected |
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components, one bounded (the interior of the polygon) and one |
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unbounded. This result is the Jordan curve theorem for polygons. |
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Moise proves this directly in \cite{moise}, pp. 16-18. |
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%A direct proof can be |
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%found in \cite{moise}, pp. 16--18. |
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\item In complex analysis, the Schwarz-Christoffel transformation |
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\cite{silverman} gives a conformal map from any polygon to the upper |
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half plane. |
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\item The area of a lattice polygon can be calculated using Pick's |
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theorem. |
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\end{enumerate} |
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The \emph{perimeter} of a polygon is the sum of the lengths of the |
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line segments in its bounding polygonal curve. |
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Some authors do not include the interior of a polygon as part of the |
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polygon, and thus identify a polygon with what we call a polygonal |
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curve. Such authors sometimes remove the requirement that the path |
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determining a polygon be simple. |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{borsuk-szmielew} |
\bibitem{borsuk-szmielew} |
| K. Borsuk and W. Szmielew, \emph{Foundations of Geometry}, |
K. Borsuk and W. Szmielew, \emph{Foundations of Geometry}, |
| North-Holland Publishing Company, 1960. |
North-Holland Publishing Company, 1960. |
| \bibitem{forder} |
\bibitem{forder} |
| H.G. Forder, \emph{The Foundations of Euclidean Geometry}, |
H.G. Forder, \emph{The Foundations of Euclidean Geometry}, |
| Dover Publications, 1958. |
Dover Publications, 1958. |
| \bibitem{moise} |
\bibitem{moise} |
| E.E. Moise, \emph{Geometric Topology in Dimensions 2 and 3}, |
E.E. Moise, \emph{Geometric Topology in Dimensions 2 and 3}, |
| Springer-Verlag, 1977. |
Springer-Verlag, 1977. |
| |
\bibitem{silverman} |
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R.A. Silverman, \emph{Introductory Complex Analysis}, |
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Dover Publications, 1972. |
| \end{thebibliography} |
\end{thebibliography} |
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| \PMlinkescapeword{segments} |
\PMlinkescapeword{segments} |
| \PMlinkescapeword{maximal} |
\PMlinkescapeword{maximal} |
| \PMlinkescapeword{name} |
\PMlinkescapeword{name} |
| \PMlinkescapeword{names} |
\PMlinkescapeword{names} |
| \PMlinkescapeword{meet} |
\PMlinkescapeword{meet} |
| \PMlinkescapeword{opens} |
\PMlinkescapeword{opens} |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
| \PMlinkescapeword{divides} |
\PMlinkescapeword{divides} |
| \PMlinkescapeword{bounded} |
\PMlinkescapeword{bounded} |
| \PMlinkescapeword{unbounded} |
\PMlinkescapeword{unbounded} |
| \PMlinkescapeword{complex} |
\PMlinkescapeword{complex} |
