<p>I think(?) I [Joe O'Rourke] was the first to introduce <em>monotone mountains</em>, but perhaps I copied
that class from someone else.  It is a subclass of monotone polygons, with one of the
two chains that constitutes a monotone polygon a single segment.
I used it in the paper,
"<a href="http://cs.smith.edu/~orourke/papers.php#10" rel="nofollow noreferrer">Vertex pi-lights for monotone mountains</a>,"
<em>9th Canad. Conf. Comput. Geom.</em>, Aug. 1997, pp. 1-5.
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img src="https://i.sstatic.net/QFpDk.jpg" alt="enter image description here">
<br />
Joe Mitchell subsequently used this class of polygons in his abstract,
"Triangulating Monotone Mountains,"
AMS 345/CSE 355 (<a href="http://www.ams.sunysb.edu/~jsbm/courses/345/triangulating-monotone-mountains.pdf" rel="nofollow noreferrer">PDF link</a>).</p>

CC BY-SA 3.0
2011-12-10T02:09:55.333Z
2011-12-12T09:42:35.730Z
2011-12-12T09:42:35.730Z
Joseph O'Rourke
<p>Godfried reports that his hierarchy paper (cited by Aaron) never got written, but some of its ideas
appeared in these two papers.</p>

<ul>
<li><p>Hossam A. ElGindy, Godfried T. Toussaint. "On geodesic properties of polygons<br>
relevant to linear time triangulation," <em>The Visual Computer</em>.                     </p></li>
<li><p>Godfried T. Toussaint, "Movable separability of sets," in <em>Computational<br>
Geometry</em>, Ed., G. T. Toussaint, North-Holland, 1985. </p></li>
</ul>

<p>Here is a figure from the first:
<br />&nbsp;&nbsp;&nbsp;<img src="https://i.sstatic.net/qKKHa.jpg" alt="Hierarchy"><br />
Note that the shapes mentioned in Yoshio Okamoto's post appear in this hierarchy.
And if you are wondering what a (Barbados-induced?) "palm" polygon looks like...
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img src="https://i.sstatic.net/KnVO5.jpg" alt="Palm Polygon">
<br /></p>

CC BY-SA 3.0
2011-12-11T14:37:22.000Z
2011-12-11T14:45:10.457Z
2011-12-11T14:45:10.457Z
Joseph O'Rourke
<p>Maybe it would make sense to make this question community wiki and solicit answers that describe additional individual classes of polygons, since a proper survey seems to be lacking (although I did run across some references to a manuscript by Toussaint)?</p>

<p>If so, here's another one: <a href="http://en.wikipedia.org/wiki/Pseudotriangle">pseudotriangles</a>, polygons that have only three convex vertices and all the rest concave.</p>

<p>Also, O'Rourke's book "Art Gallery Theorems and Algorithms" has a section on "spiral polygons", polygons whose concave vertices form a single contiguous subsequence of the boundary.</p>

CC BY-SA 3.0
2011-12-09T12:38:31.693Z
Hazel Everett wrote her Ph.D. thesis (under Derek Corneil) on spiral polygons: "Recognizing Visibility Graphs of Spiral Polygons," _J. Algorithms_, 1990: 1-26.
Joseph O'Rourke
CC BY-SA 3.0
2011-12-09T01:08:27.707Z
2011-12-09T01:08:27.707Z
David Eppstein
<p>Another class of polygons is called anthropomorphic: these polygons have exactly two ears and one mouth. </p>

<p>Godfried T.Toussaint, "Anthropomorphic Polygons." American Mathematical Monthly 122, 31-35, 1991.</p>

<p>Grunbaum wrote a long paper describing many classes of crossing polygons here:
Branko Grunbaum, "Polygons," In The Geometry of Metric and Linear Spaces, L. M. Kelly, ed. Lecture Notes in Mathematics Number 490, pp. 147 - 184. Springer-Verlag, Berlin-Heidelberg-New York 1975.</p>

CC BY-SA 3.0
2011-12-11T15:47:53.943Z
2011-12-11T15:47:53.943Z
Godfried Toussaint
<p>"Two-Convex Polygons,"
O. Aichholzer,  F. Aurenhammer,
F. Hurtado,
P.A. Ramos,
J. Urrutia, 2009.</p>

<p><br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="https://i.sstatic.net/gXnGc.jpg" alt="Two-convex"><br /></p>

<blockquote>
  <p>A polygon is $k$-convex if and only if its stabbing number is at most $2k$.</p>
</blockquote>

CC BY-SA 3.0
2011-12-11T14:29:30.097Z
2011-12-11T14:29:30.097Z
Joseph O'Rourke
<p>This is not a serious answer, just an opportunity to mention that Godfried Toussaint
introduced <em>sail polygons</em> in his 1985 paper "<a href="http://www.springerlink.com/content/q744325k7350m237/" rel="nofollow noreferrer">A simple linear algorithm for intersecting convex polygons</a>":
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img src="https://i.sstatic.net/LVSpo.jpg" alt="Sail Polygon"><br />
I don't think this particular class found many subsequent uses. :-)</p>

CC BY-SA 3.0
2011-12-08T20:16:56.527Z
There is an unpublished manuscript of Toussaint that is supposed to have a collection of polygons, "A hierarchy of simple polygons." I am unable to find it online. Do you know if it ever appeared in any form?
Aaron Sterling
CC BY-SA 3.0
2011-12-08T23:13:08.623Z
And then what do we call the union of a human polygon and a sail polygon ? :)
Suresh Venkat
CC BY-SA 3.0
2011-12-09T14:25:55.690Z
@Aaron: I have a query in to Godfried, and will let you know eventually.
Joseph O'Rourke
CC BY-SA 3.0
2011-12-08T20:03:03.803Z
2011-12-08T20:03:03.803Z
Joseph O'Rourke
<p>The following paper by Hernando, Houle, and Hurtado introduces a few classes of simple polygons in terms of visibility.</p>

<ul>
<li>M. Carmen Hernando, Michael E. Houle, Ferran Hurtado: On local transformation of polygons with visibility properties. Theor. Comput. Sci. 289(2): 919-937 (2002) <a href="http://dx.doi.org/10.1016/S0304-3975(01)00409-1" rel="nofollow">http://dx.doi.org/10.1016/S0304-3975(01)00409-1</a></li>
</ul>

<p>Now I describe the definitions of their polygons.</p>

<p><strong>Edge-visible polygons</strong></p>

<p>Let $e$ be an edge of a simple polygon $P$.  Then, $P$ is <em>edge-visible</em> from $e$ if for every point $p$ of $P$ there exists a point $q$ on the relative interior of $e$ such that the relative interior of the line segment connecting $p$ and $q$ completely lies outside of $P$.</p>

<p><strong>Externally visible polygons</strong></p>

<p>A simple polygon is <em>externally visible</em> if for every point $p$ on the boundary of $P$ there exists a ray (half line) $r$ starting at $p$ that intersects $P$ only at $p$.</p>

<p>Monotone polygons, star-shaped polygons and pseudo-triangles are externally visible.</p>

CC BY-SA 3.0
2011-12-11T02:06:21.123Z
2011-12-11T02:06:21.123Z
Yoshio Okamoto
<p><strong>UPDATE</strong></p>

<p>Now community wiki.  My new version of the question is: let's make a big list of classes of polygons.  We may be able to produce the most comprehensive list on the web, or in the literature. If there is community interest, after Jan 1st I will organize the information from all the answers into a post on the community blog.</p>

<p><strong>ORIGINAL QUESTION BELOW</strong></p>

<blockquote>
  <p>Could you recommend a source, either in print or online, for a menagerie of polygons?  An extensive/exhaustive list of classes of polygon?</p>
</blockquote>

<p>The Wikikpedia article on polygons provides a <a href="http://en.wikipedia.org/wiki/Polygon#Classification" rel="nofollow">partial classification</a>, but I would like something more complete.  Also, I am not concerned about how to name a 90-sided figure.  Rather, I am trying to find a list of classes that contain infinitely many figures each (examples: star polygon, isothetic polygon).  </p>

<p>For example: actual words for "looks like a spider," "would look like a football if it were smoothed out," and so on.</p>

<p>I've already checked <a href="http://www.ics.uci.edu/~eppstein/junkyard/" rel="nofollow">The Geometry Junkyard</a> and several pages of a Google search, but perhaps I didn't know what to look for.</p>

<p>Thanks very much.</p>

CC BY-SA 3.0
2011-12-07T20:40:20.920Z
For those of us who like me are not native English speakers: [here's a definition of a menagerie](http://dictionary.reference.com/browse/menagerie)
Anthony Labarre
CC BY-SA 3.0
2011-12-07T22:25:46.637Z
There's a small listing in Subhash Suri's survey "Polygons" in the Handbook of Discrete and Computational Geometry. But I hesitate to make this an answer rather than a comment, because it's less complete than the Wikipedia article you already list. It lists only simple, convex, monotone, star-shaped, and orthogonal polygons. See also http://www.staff.science.uu.nl/~kreve101/papers/personpoly.pdf for some more obscure types of polygon.
David Eppstein
CC BY-SA 3.0
2011-12-08T04:23:28.743Z
The paper cited by @DavidEppstein is worth it just for its observation that distinguishing "the male person polygons from the female person polygons" is an open problem that "may be easier to solve in 3-dimensional space than in the plane."  Priceless.
mjqxxxx
CC BY-SA 3.0
2011-12-08T19:15:35.743Z
@DavidEppstein:Thanks for the pointers, but, more importantly, CONGRATULATIONS on becoming an ACM Fellow. :-)
Aaron Sterling
CC BY-SA 3.0
2011-12-09T04:57:36.370Z
Aaron, what do you think about @DavidEppstein's suggestion to make this CW ?
Suresh Venkat
CC BY-SA 3.0
2011-12-09T15:05:22.333Z
A paper by Hernando, Houle, and Hurtado (TCS 2002) introduced a few classes of simple polygons (http://dx.doi.org/10.1016/S0304-3975(01)00409-1). I'd turn this reference to an answer when this question is made community wiki.
Yoshio Okamoto
CC BY-SA 3.0
2011-12-09T15:20:11.680Z
@SureshVenkat: I would support the suggestion; please change this to CW.  I was initially thinking about starting a CW answer to collect everything, but if there are different answers, it would be easier to post diagrams, as Joseph O'Rourke did.  If the CW generates a lot of interest, I could collect the information into a community blog post in the new year.
Aaron Sterling
<cg.comp-geom><big-list><polygon>
Menagerie of polygons
Aaron Sterling
# Menagerie of polygons

(Asked by: username_0 on 2011-12-07)

**UPDATE**

Now community wiki. My new version of the question is: let's make a big list of classes of polygons. We may be able to produce the most comprehensive list on the web, or in the literature. If there is community interest, after Jan 1st I will organize the information from all the answers into a post on the community blog.

**ORIGINAL QUESTION BELOW**

> Could you recommend a source, either in print or online, for a menagerie of polygons? An extensive/exhaustive list of classes of polygon?

The Wikikpedia article on polygons provides a [partial classification](http://en.wikipedia.org/wiki/Polygon#Classification), but I would like something more complete. Also, I am not concerned about how to name a 90-sided figure. Rather, I am trying to find a list of classes that contain infinitely many figures each (examples: star polygon, isothetic polygon).

For example: actual words for "looks like a spider," "would look like a football if it were smoothed out," and so on.

I've already checked [The Geometry Junkyard](http://www.ics.uci.edu/~eppstein/junkyard/) and several pages of a Google search, but perhaps I didn't know what to look for.

Thanks very much.

**Tags:** cg.comp-geom, big-list, polygon
> **Comments:**
> **username_1**: For those of us who like me are not native English speakers: [here's a definition of a menagerie](http://dictionary.reference.com/browse/menagerie)  
> **username_2**: There's a small listing in Subhash Suri's survey "Polygons" in the Handbook of Discrete and Computational Geometry. But I hesitate to make this an answer rather than a comment, because it's less complete than the Wikipedia article you already list. It lists only simple, convex, monotone, star-shaped, and orthogonal polygons. See also http://www.staff.science.uu.nl/~kreve101/papers/personpoly.pdf for some more obscure types of polygon.  
> **username_3**: The paper cited by @DavidEppstein is worth it just for its observation that distinguishing "the male person polygons from the female person polygons" is an open problem that "may be easier to solve in 3-dimensional space than in the plane."  Priceless.  
> **username_0**: @DavidEppstein:Thanks for the pointers, but, more importantly, CONGRATULATIONS on becoming an ACM Fellow. :-)  
> **username_4**: Aaron, what do you think about @DavidEppstein's suggestion to make this CW ?  
> **username_5**: A paper by Hernando, Houle, and Hurtado (TCS 2002) introduced a few classes of simple polygons (http://dx.doi.org/10.1016/S0304-3975(01)00409-1). I'd turn this reference to an answer when this question is made community wiki.  
> **username_0**: @SureshVenkat: I would support the suggestion; please change this to CW.  I was initially thinking about starting a CW answer to collect everything, but if there are different answers, it would be easier to post diagrams, as Joseph O'Rourke did.  If the CW generates a lot of interest, I could collect the information into a community blog post in the new year.  

---

## 7 answers

### Answer by username_6 on 2011-12-10. Score: 6

I think(?) I [Joe O'Rourke] was the first to introduce *monotone mountains*, but perhaps I copied
that class from someone else. It is a subclass of monotone polygons, with one of the
two chains that constitutes a monotone polygon a single segment.
I used it in the paper,
"[Vertex pi-lights for monotone mountains](http://cs.smith.edu/~orourke/papers.php#10),"
*9th Canad. Conf. Comput. Geom.*, Aug. 1997, pp. 1-5.
  
     ![enter image description here](https://i.sstatic.net/QFpDk.jpg)
  
Joe Mitchell subsequently used this class of polygons in his abstract,
"Triangulating Monotone Mountains,"
AMS 345/CSE 355 ([PDF link](http://www.ams.sunysb.edu/~jsbm/courses/345/triangulating-monotone-mountains.pdf)).


---

### Answer by username_6 on 2011-12-11. Score: 6

Godfried reports that his hierarchy paper (cited by Aaron) never got written, but some of its ideas
appeared in these two papers.

* Hossam A. ElGindy, Godfried T. Toussaint. "On geodesic properties of polygons  
  relevant to linear time triangulation," *The Visual Computer*.
* Godfried T. Toussaint, "Movable separability of sets," in *Computational  
  Geometry*, Ed., G. T. Toussaint, North-Holland, 1985.

Here is a figure from the first:


---

### Answer by username_2 on 2011-12-09. Score: 5

Maybe it would make sense to make this question community wiki and solicit answers that describe additional individual classes of polygons, since a proper survey seems to be lacking (although I did run across some references to a manuscript by Toussaint)?

If so, here's another one: [pseudotriangles](http://en.wikipedia.org/wiki/Pseudotriangle), polygons that have only three convex vertices and all the rest concave.

Also, O'Rourke's book "Art Gallery Theorems and Algorithms" has a section on "spiral polygons", polygons whose concave vertices form a single contiguous subsequence of the boundary.

> **Comments:**
> **username_6**: Hazel Everett wrote her Ph.D. thesis (under Derek Corneil) on spiral polygons: "Recognizing Visibility Graphs of Spiral Polygons," _J. Algorithms_, 1990: 1-26.  

---

### Answer by username_7 on 2011-12-11. Score: 5

Another class of polygons is called anthropomorphic: these polygons have exactly two ears and one mouth.

Godfried T.Toussaint, "Anthropomorphic Polygons." American Mathematical Monthly 122, 31-35, 1991.

Grunbaum wrote a long paper describing many classes of crossing polygons here:
Branko Grunbaum, "Polygons," In The Geometry of Metric and Linear Spaces, L. M. Kelly, ed. Lecture Notes in Mathematics Number 490, pp. 147 - 184. Springer-Verlag, Berlin-Heidelberg-New York 1975.


---

### Answer by username_6 on 2011-12-11. Score: 5

"Two-Convex Polygons,"
O. Aichholzer, F. Aurenhammer,
F. Hurtado,
P.A. Ramos,
J. Urrutia, 2009.

          
![Two-convex](https://i.sstatic.net/gXnGc.jpg)

> A polygon is $k$-convex if and only if its stabbing number is at most $2k$.


---

### Answer by username_6 on 2011-12-08. Score: 3

This is not a serious answer, just an opportunity to mention that Godfried Toussaint
introduced *sail polygons* in his 1985 paper "[A simple linear algorithm for intersecting convex polygons](http://www.springerlink.com/content/q744325k7350m237/)":
  
      ![Sail Polygon](https://i.sstatic.net/LVSpo.jpg)  
I don't think this particular class found many subsequent uses. :-)

> **Comments:**
> **username_0**: There is an unpublished manuscript of Toussaint that is supposed to have a collection of polygons, "A hierarchy of simple polygons." I am unable to find it online. Do you know if it ever appeared in any form?  
> **username_4**: And then what do we call the union of a human polygon and a sail polygon ? :)  
> **username_6**: @Aaron: I have a query in to Godfried, and will let you know eventually.  

---

### Answer by username_5 on 2011-12-11. Score: 3

The following paper by Hernando, Houle, and Hurtado introduces a few classes of simple polygons in terms of visibility.

* M. Carmen Hernando, Michael E. Houle, Ferran Hurtado: On local transformation of polygons with visibility properties. Theor. Comput. Sci. 289(2): 919-937 (2002) <http://dx.doi.org/10.1016/S0304-3975(01)00409-1>

Now I describe the definitions of their polygons.

**Edge-visible polygons**

Let $e$ be an edge of a simple polygon $P$. Then, $P$ is *edge-visible* from $e$ if for every point $p$ of $P$ there exists a point $q$ on the relative interior of $e$ such that the relative interior of the line segment connecting $p$ and $q$ completely lies outside of $P$.

**Externally visible polygons**

A simple polygon is *externally visible* if for every point $p$ on the boundary of $P$ there exists a ray (half line) $r$ starting at $p$ that intersects $P$ only at $p$.

Monotone polygons, star-shaped polygons and pseudo-triangles are externally visible.


---

