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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeMay 5th 2021
   • (edited May 5th 2021)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexA __polygonal line__ is the union of segments $\overline{A_0 A_1}$, $\overline{A_1 A_2}, \ldots, \overline{A_{n-1},A_n}$ where $n\in\mathbf{N}$ and $(A_0,A_1,\ldots, A_n)$ is an $(n+1)$-tuple of consecutively distinct points in an affine plane or in a $k$-dimensional [[affine space]]. A polygonal line is called __closed__ if $A_0 = A_n$ (regarding that consecutive points are distinct that implies that $n\geq 2$). A closed polygonal line is __non-self-intersecting__ if for any $k \lt l$, $0\leq k\lt l \leq n-1$, $\overline{A_k A_{k+1}}\cap \overline{A_l A_{l+1}}= \emptyset$ except in the case $(k,l) = (0,n-1)$ when the intersection is $\{A_0\}$. A __simple (planar) polygonal line__ is a closed non-self-intersecting line which is in its entirety contained in some affine 2-plane. By Jordan's curve theorem, a complement of any simple polygonal line within its 2-plane has two components, one bounded and one unbounded. A __polygon__ is a set in an affine $k$-space ($k\geq 2$) which can be represented as a simple closed planar polygonal line union the bounded component of its complement within its 2-plane. The bounded and unbounded components of the complement of the corresponding polygonal line in its 2-plane are called the interior and the exterior of the polygon. For fixed $n$ we also say $n$-gon, and the points $A_1,\ldots, A_n$ are called the vertices of the polygon, the segments $\overline{A_n A_1}$, $\overline{A_1 A_2}, \ldots, \overline{A_{n-1},A_n}$ are called the edges or the sides of the polygon. A 3-gon is also called a triangle and it may be also defined as the smallest convex set containing 3 non-colinear points called the vertices of the triangle. $n$-gons for $n\geq 4$ may be convex or nonconvex. In other geometries instead of segments we may consider segments on geodesical lines, so we can talk about $n$-gones in spherical and non-Euclidean geometries. For some related ideas see [[polyhedron]]. <a href="https://ncatlab.org/nlab/revision/diff/polygon/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/polygon/7">v7</a>, <a href="https://ncatlab.org/nlab/show/polygon">current</a>

   A polygonal line is the union of segments $\overline{A_0 A_1}$, $\overline{A_1 A_2}, \ldots, \overline{A_{n-1},A_n}$ where $n\in\mathbf{N}$ and $(A_0,A_1,\ldots, A_n)$ is an $(n+1)$-tuple of consecutively distinct points in an affine plane or in a $k$-dimensional affine space. A polygonal line is called closed if $A_0 = A_n$ (regarding that consecutive points are distinct that implies that $n\geq 2$). A closed polygonal line is non-self-intersecting if for any $k \lt l$, $0\leq k\lt l \leq n-1$, $\overline{A_k A_{k+1}}\cap \overline{A_l A_{l+1}}= \emptyset$ except in the case $(k,l) = (0,n-1)$ when the intersection is $\{A_0\}$. A simple (planar) polygonal line is a closed non-self-intersecting line which is in its entirety contained in some affine 2-plane. By Jordan’s curve theorem, a complement of any simple polygonal line within its 2-plane has two components, one bounded and one unbounded. A polygon is a set in an affine $k$-space ($k\geq 2$) which can be represented as a simple closed planar polygonal line union the bounded component of its complement within its 2-plane. The bounded and unbounded components of the complement of the corresponding polygonal line in its 2-plane are called the interior and the exterior of the polygon. For fixed $n$ we also say $n$-gon, and the points $A_1,\ldots, A_n$ are called the vertices of the polygon, the segments $\overline{A_n A_1}$, $\overline{A_1 A_2}, \ldots, \overline{A_{n-1},A_n}$ are called the edges or the sides of the polygon. A 3-gon is also called a triangle and it may be also defined as the smallest convex set containing 3 non-colinear points called the vertices of the triangle. $n$-gons for $n\geq 4$ may be convex or nonconvex.

   In other geometries instead of segments we may consider segments on geodesical lines, so we can talk about $n$-gones in spherical and non-Euclidean geometries. For some related ideas see polyhedron.

   diff, v7, current

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeMay 5th 2021
   • (edited May 5th 2021)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI added some discussion pointing out that it is not trivial to avoid the (polygonal case of) Jordan's curve theorem to discuss the interior. <a href="https://ncatlab.org/nlab/revision/diff/polygon/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/polygon/7">v7</a>, <a href="https://ncatlab.org/nlab/show/polygon">current</a> P.S. An interesting reference on the polygonal case of the Jordan's curve theorem is <https://jeffe.cs.illinois.edu/teaching/comptop/2009/notes/jordan-polygon-theorem.pdf>

   I added some discussion pointing out that it is not trivial to avoid the (polygonal case of) Jordan’s curve theorem to discuss the interior.

   diff, v7, current

   P.S. An interesting reference on the polygonal case of the Jordan’s curve theorem is https://jeffe.cs.illinois.edu/teaching/comptop/2009/notes/jordan-polygon-theorem.pdf

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 6th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the link to *[[Jordan's curve theorem]]* <a href="https://ncatlab.org/nlab/revision/diff/polygon/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/polygon/8">v8</a>, <a href="https://ncatlab.org/nlab/show/polygon">current</a>

   added the link to Jordan’s curve theorem

   diff, v8, current