polygon

1 Definitions

We follow Forder [2] for most of this entry.The term polygon can be defined if one has a definition of an interval. For thisentry we use betweenness geometry. A betweenness geometryis just one for which there is a set of points and a betweenness relation $B$ defined.Rather than write $(a,b,c)\in B$ we write $a*b*c$.

1. If $a$ and $b$ are distinct points, the line $a\mathit{}b$ is the set ofall points $p$ such that $p*a*b$ or $a*p*b$ or $a*b*p$. It can be shownthat the line $ab$ and the line $ba$ are the same set of points.

2. If $o$ and $a$ are distinct points, a ray $\mathrm{[}oa$ is the set of all points $p$ such that$p=o$ or $o*p*a$ or $o*a*p$.

3. If $a$ and $b$ are distinct points, the open interval is the set of points$p$ such that $a*p*b$. It is denoted by $(a,b).$

4. If $a$ and $b$ are distinct points, the closed interval is$(a,b)\cup \{a\}\cup \{b\}$, and denoted by $[a,b].$

5. The way $a_{\mathrm{1}}\mathit{}{a}_{\mathrm{2}}\mathit{}\mathrm{\dots }\mathit{}{a}_n$ is the finite set of points $\{a_1,\mathrm{\dots },a_n\}$along with the open intervals $(a_1,a_2),(a_2,a_3),\mathrm{\dots },(a_{n-1},a_n)$.The points $a_1,\mathrm{\dots },a_n$ are called the vertices of the way, and theopen intervals are called the sides of the way.A way is also called a broken line.The closed intervals $[a_1,a_2],\mathrm{\dots },[a_{n-1},a_n]$ are called the side-intervals ofthe way. The lines $a_1{a}_2,\mathrm{\dots },a_{n-1}{a}_n$ are called the side-linesof the way.The way $a_1{a}_2\mathrm{\dots }{a}_n$ is said to join $a_1$ to $a_n$.It is assumed that $a_{i-1},a_i,a_{i+1}$ are not collinear.

6. A way is said to be simple if it does not meet itself. To be precise,(i) no two side-intervals meet in any point which is not a vertex, and (ii) no three side-intervalsmeet in any point.

7. A polygon is a way $a_1{a}_2\mathrm{\dots }{a}_n$ for which $a_1=a_n$. Notice that there isno assumption that the points are coplanar.

8. A simple polygon is polygon for which the way is simple.

9. A region is a set of points not all collinear, any two of which can be joined by points of a way usingonly points of the region.

10. A region $R$ is convex if for each pair of points $a,b\in R$ the open interval $(a,b)$ iscontained in $R.$

11. Let $X$ and $Y$ be two sets of points. If there is a set of points $S$ such that every wayjoining a point of $X$ to a point of $Y$ meets $S$ then $S$ is said to separate$X$ from $Y$.

12. If $a_1{a}_2\mathrm{\dots }{a}_n$ is a polygon, then the angles of the polygon are$\mathrm{\angle }{a}_n{a}_1{a}_2,\mathrm{\angle }{a}_1{a}_2{a}_3$, and so on.

Now assume that all points of the geometry are in one plane. Let $P$ be a polygon. ($P$ is calleda plane polygon.)

1. A ray or line which does not go through a vertex of $P$ will be called suitable.

2. An inside point $a$ of $P$ is one for which a suitable ray from $a$meets $P$ an odd number of times. Points that are not on or inside $P$ are said to be outside$P$.

3. Let $\{P_i\}$ be a set of polygons. We say that $\{P_i\}$ dissect $P$ if the followingthree conditions are satisfied: (i) $P_i$ and $P_j$ do not have a common inside point for $i\ne j$,(ii) each inside point of $P$ is inside or on some $P_i$ and (iii) each inside point of $P_i$ isinside $P$.

4. A convex polygon is one whose inside points are all on the same side of any side-lineof the polygon.

2 Theorems

Assume that all points are in one plane. Let $P$ be a polygon.

1. It can be shown that $P$ separates the other points of the plane into at least two regions and thatif $P$ is simple there are exactly two regions. Moise proves this directly in [3], pp. 16-18.

2. It can be shown that $P$ can be dissected into triangles $\{T_i\}$ such thatevery vertex of a $T_i$ is a vertex of $P$.

3. The following theorem of Euler can be shown: Suppose $P$ is dissected into $f>1$ polygonsand that the total number of vertices of these polygons is $v$, and the number of open intervalswhich are sides is $e$. Then

   $$v-e+f=1$$

   .

A plane simple polygon with $n$ sides is called an $n$-gon, although for small $n$there are more traditional names:

A plane simple polygon is also called a Jordan polygon.