The polygons tag has no usage guidance.

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### Geodesic Digons in Reductive Spaces

Consider a naturally reductive homogeneous space $M$ of positive curvature. Is it necessarily the case that there exists a geodesic digon whose interior angles sum to less than $2\pi$? Angles are ...

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### Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?
You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...

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### Gauss-Wantzel theorem, Fermat primes and solvability of S_n [closed]

Gauss-Wantzel theorem asserts that a polygon with $n$ sides is constructible if and only if $n$ is a product of a power of $2$ and distinct prime Fermat numbers, where the Fermat number of index $k$ ...

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### Terminology for polygons

As you may know term "polygon" might mean few different things
and its meaning has to guessed from context.
By some reason I have to use few of these meaning in one place.
So I converge to the ...

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### Detect perimetral edges of a polygon [closed]

I'm developing a building editor. Users can draw rooms by adding angles (vertices of the room) with a left click. Clicking on an existing angle closes the room and fills the floor by using the ...

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### Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?

Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within ...

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### Construct polygon/polyhedron containing all points not externally visible w.r.t given polygon/polyhedron?

Is there an algorithm to construct a polyhedron containing all points in space for which there exists no ray to infinite not intersecting a given polyhedron?
In 2D, we could consider polygons. For ...

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### Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...

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### Which polygons have *simple* periodic billiard paths?

I know (or, rather, believe) that it remains unknown whether every polygon
has a periodic billiard path.
But Masur proved in the 1980's that every rational polygon
(vertex angles rational multiples ...

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### Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?

Not sure whether this question belongs here or math.stackexchange.
You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...

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### Measuring the Randomness and Statistics of Convex Polygons

How can I tell, how likely it is, that a given convex polygon with a sufficiently high number of edges is random and, if so, what kind of randomness it is (e.g. white noise)?
What is known about ...

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### Calculating the “Belvedere Hull” of a Simple Planar Polygon

As an informal motivation the problem, imagine a tower with polygonal footprint, that is located in a beautiful landscape, the "Belvedere Hull" is then related to the directions, in which one would ...

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359 views

### Does the boundary of a convex body contain a regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ includes 5 points which form a regular planar pentagon? The following consideration suggests the answer "yes": if we ...

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### What does this number tell me about a convex lattice polygon?

EDIT: I realized I'd tricked myself by working with a too special case of $f$, the question is now updated (boundary lattice points replaced vertices).
Suppose I have a convex lattice polygon $P$, ...

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### Determine the boundary points of a set of points [closed]

I have a set of points $S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$. Then how to find the boundary points (which is a subset of $S$) of $S$?
There are methods like convex hull, concave hull and ...

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### Intersection points of closed curves inscribed in a convex polygon

Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...

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### Non-Convex Polygons with “Antipodal Visibility”

by "antipodal visibility" of planar, simple polygons I mean the following property:
if two points $p$ and $q$ on the polygon's boundary divide the polygon's boundary into two polylines of equal ...

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### What polygons can be shrunk into themselves?

Let's call a polygon $P$ shrinkable if any down-scaled (dilated) version of $P$ can be translated into $P$. For example, the following triangle is shrinkable (the original polygon is green, the ...

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### Most Regularity of a Polygon

Conseider $n$ electrons in an empty sphere. What structure do they make?
This question have two cases: (i) if electrons should be sit on the boundry of sphere (one can suppose that the boundry of ...

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### Find shift direction for min overlap area of 2 polygons

I have 2 arbitrary polygons (concave or convex) with certain overlap.
Now there is some relative shift between these 2 polygons (vector s with a constant length).
I want to find the direction of s ...

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### Shrink polygon to a specific area by offsetting

I have a 2D polygon that I want to shrink by a specific offset (A) to match a certain area ratio (R) of the original polygon. Is there a formula or algorithm for such a problem? I am interested in a ...

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### What is the median area of a random n-gon inside a unit square?

A square is bounded by the coordinates (0,0), (0,1), (1,0) and (1,1). Random x and y coordinates are chosen in the interval [0,1] for each of the n points. The n points are then randomly connected to ...

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### What is the shape of the convex $n$ -gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $B_n$? ...

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### What is the shape of the $n$-gon which gives the maximum of a function?

What is the shape of the $n$-gon $P_1P_2\cdots P_n$ which gives the maximum of $A_n$? The quantity $A_n$ is defined by
$$ A_n = \frac{{\sum_{i\lt{j}\le{n}}{\lvert P_i ...

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### Algorithms for covering a rectilinear polygon using the same multiple rectangles

Sorry for the crossing-posting: original post is here
All angles of the polygon (representing a room) are right. It may be convex or concave. Use rectangles of the same size (representing a sensor ...

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### Determining A valid Convex Hexagon given The length of Six sides

Suppose We are given the length of all six sides of a Convex Hexagon. How can we tell whether it's valid or Not ? that means can we tell whether it's area is positive or not ?

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### How to find overlap between two convex hulls,along with the overlap area

I have two boundaries of two planar polygons , say,B1 and B2 of polygons P1 and P2(with m and n points in Boundaries B1 and B2). I want to find out if the Polygons overlap or not.If they overlap,then ...

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### cyclic polygons & trigonometry

I posted this question to stackexchange, where it's generated some comments but no progress toward answering it. I'm going to say somewhat more here than I did there.
At one vertex of a pentagon ...

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### Determine if you can build a polygon from segments [closed]

Is there a way to determine whether it is possible to build a polygon from given n segments?
Maybe triangle inequality generalized?

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### Polar interpretation of convexity

Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...

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### Quadrilateral from 4 random points

Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points?
I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q.
But ...

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### Partitioning a polygon into convex parts

I'm looking for an algorithm to partition any simple closed polygon into convex sub-polygons--preferably as few as possible.
I know almost nothing about this subject, so I've been searching on Google ...

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### Get a point inside a polygon

I have a 2D polygon of arbitrary geometry. I need to find any point that is inside of that polygon. Taking the center won't work, because the polygon might not be convex. Is there a way to quickly ...

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### Using mirrors to make a non-convex polygon visible from a fixed interior point

Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...

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### Is there a simple criterion to determine if two parallelograms intersect?

Assume we are given two parallelograms in the plane. How can I check if their intersection is nonempty?
Note that I do not need to actually find the intersection.

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### Elementary problem about triangles inside a convex polygon

Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...

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### Nonconvex manhole covers

One common reason given for the circularity of manhole covers is that they can't fall through the manhole. For convex manhole covers, this property is equivalent to having constant width — if ...

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### Compute the Centroid of a 3D Planar Polygon

Given a list of 3D coordinates that define the surface( Point3D1, Point3D2, Point3D3, and so ...

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### Dividing a square into 5 equal squares

Can you divide one square paper into five equal squares?
You have a scissor and glue. You can measure and cut and then attach as well. Only condition is You can't waste any paper.

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### Heaviest Convex Polygon

Suppose we have an arbitrary function $f : \mathbb{R}^2 \to \mathbb{R}$. For any subset $s \subseteq \mathbb{R}^2$, we can define $g_f(s)$ as the integral* of $f$ over the region $s$. Suppose ...