# Questions tagged [polygons]

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70
questions

**3**

votes

**0**answers

154 views

### Hypothesis: An injection from polygons into $SO(2) \times S_n$

I have stumbled upon a possible representation of polygons by a concise description of their behaviour under rotation. I would like to know more about it and, obviously, if it is actually a bijection. ...

**9**

votes

**2**answers

511 views

### Strange formula for area of a convex polygon

Consider a convex $n-$gon in $\mathbb{R}^2$ with sides contained in the lines $y=k_ix+b_i, 1\leq i\leq n.$ Then its area equals to
$$
S=\frac{1}{2}\sum_{i=1}^{n} \frac{(b_{i+1}-b_i)^2}{k_{i+1}-k_i}.
$$...

**0**

votes

**1**answer

98 views

### Point generation in polygon

I know about the Halton sequence. But so far I can’t find the formulas by which points are generated. Also worried is the question Halton sequence generates points only in the rectangle? Or can I ...

**1**

vote

**1**answer

73 views

### Can we split a polygon in half (vertex-wise) by a diagonal, but with a constant maximum difference?

This is a followup to my last question - Can we prove that simple polygons can always be split in half (vertex-wise) by diagonals?
Is there a constant natural number K for which the following is true:...

**0**

votes

**1**answer

55 views

### Can we prove that simple polygons can always be split in half (vertex-wise) by diagonals?

Can we prove that for any simple polygon with more than 3 vertices there always exists a diagonal which:
is inside the polygon
doesn't intersect with any edges
splits the polygon in two polygons in ...

**0**

votes

**0**answers

77 views

### Automorphism group of the cycle graph with $k$ diagonals

Let $C_n$ be the cycle graph on $n$ vertices. Then $Aut(C_n) \cong D_{2n}$ the dihedral group of order $n$.
Now, let $C_{n,i}$ be the cycle graph with $i$ many diagonals connecting opposite vertices ...

**2**

votes

**0**answers

42 views

### Complexity of existence of simple polygonalization with prescribed area?

This is a followup on my previous question. Fekete proved the NP-completeness of deciding the existence of simple polygonalization with minimum (or maximum) enclosed area (simple polygonalization is ...

**1**

vote

**1**answer

90 views

### Projections of particular simplex yielding boundary of a regular polygon?

What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?

**2**

votes

**1**answer

122 views

### Build reversed No-Fit-Polygon

I need some robust algorithm to optimally fit one non-convex polygon into another. The destination one can contain holes.
Recently I found scholarly articles on this subject:
One of them describes ...

**5**

votes

**1**answer

195 views

### Generalizations of the “Curious Tiger” Polygon

I actually don't know, whether the polygon I describe here already has name, but let me explain the problem, that is solved by the polygon, with a little story:
Imagine a flat terrain with bushes ...

**1**

vote

**1**answer

97 views

### Chain rotation of a point

Let $n$ be a positive integer number and $P$ be a point in a plane. Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ points in the plane, we take modulo $m$ for $A_j$ (it is mean $A_{m+i}=A_{i}$ for $i=1, 2, \...

**3**

votes

**2**answers

115 views

### Complexity of 2D-Minkowski sum of non-convex polygons

I have read that the complexity of computing the Minkowski-Sum of $2$ non-convex polygons (through convex decomposition) is $O(m^2 n^2)$, where $m$ and $n$ is the number of vertices of each polygon. ...

**1**

vote

**1**answer

154 views

### How many points appear in the plane when the chain of n-gons is close?

Let $A_{11}A_{12}\cdots A_{1n}$ be a regular $n$ polygon, we call $A_{11}A_{12}\cdots A_{1n}$ is the $1st-n-gons$. Now we construct the $2nd-n-gon$ based two condition as follows:
$2nd-n-gons$ is ...

**2**

votes

**0**answers

163 views

### In arbitrary cyclic polygon then $\sum_{i<j} x_{ij}^\alpha \ge \sum_{i<j} y_{ij}^\alpha $

I am looking for a proof of the inequality as follows:
Conjecture: Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$. Let $B_1B_2....B_n$ be a polygon incribed the circle $(O)...

**5**

votes

**2**answers

158 views

### An inequality related to area and sidelengths of a polygon $Area(A_1A_2…A_n) \le \frac{1}{n}cotg{\frac{\pi}{n}} \sum_{i=1}^nA_iA_{i+1}^2$

I am looking for a proof (or a reference) of an inequality related to a rea and the sidelengths of a polygon as follows:
Let $A_1A_2...A_n$ be arbitrary polygon, then:
$$Area(A_1A_2....A_n) \...

**4**

votes

**2**answers

288 views

### Fitting one Polygon in another

I have two Polygons A and B and I want to find the position, rotation and scale of B, so it fits into A and has the maximum Area possible. Also both can be concave.
I did some research but couldn't ...

**3**

votes

**2**answers

833 views

### What is the name of the 65537-gon? [closed]

I know the name of the heptadecagon (17 sides) and the diacosipentacontaheptagon (257 sides). But what is the name of the polygon with 65537 sides? I am unable to figure it.

**4**

votes

**1**answer

244 views

### Reordering vertices of a polygon

Let $Q,Q'$ be two planar polygons with the same number $n>3$ of vertices. There is a correspondence between vertices of $Q$ and $Q'$: to any vertex $z$ of $Q$ corresponds a unique vertex $z'$ of $Q'...

**40**

votes

**2**answers

3k views

### Can one “hear” the shape of a polygon via external reflections?

This question is a rough analog of Kac's "Can One Hear the Shape of a Drum?"
A closer analog is the recent "Bounce Theorem" that says, roughly, the shape of a polygon is determined by its billiard-...

**1**

vote

**0**answers

104 views

### Is it possible to dissect a regular polygon into mirrored-symmetric pieces?

Q1. Planar regular triangle is dissected into three congruent pieces, each of them having no symmetry axis.
Can it be so, that one of these pieces is a mirrored (and then rotated) copy of some other ...

**1**

vote

**0**answers

123 views

### Rectangular Newton polygon of a Jacobian pair

Let $p,q \in k[x,y]$, $k$ is a field of characteristic zero.
By definition, $p,q$ is a Jacobian pair if their Jacobian is invertible in $k[x,y]$, namely, $p_xq_y-p_yq_x \in k^*$, and $p,q$ is an ...

**16**

votes

**2**answers

792 views

### Maximum area of the intersection of a parallelogram and a triangle

How large can the intersection of a parallelogram (or a square, if you prefer) with a triangle be, if each of them is of unit area? It is easy to see that the intersection can be of area 3/4 – is this ...

**6**

votes

**2**answers

273 views

### Problem on distances in a polygon

In $\mathbb{R}^2$ consider a square (call it $S$) and three triangles (one acute $T_2$ and two obtuse $T_1$ and $T_3$) such that each triangle shares one different side with the square and the ...

**4**

votes

**1**answer

891 views

### decompose rectilinear polygons into rectangles with minimum internal borders

I have a number of rectangles as shown in the attached diagram (labelled original) and I want to merge them together in a way (labelled Result ) to get the minimum internal borders. The reason I want ...

**1**

vote

**1**answer

568 views

### Algo for covering maximum surface of a polygon with rectangles

I'm looking to an algorithm to covering maximum surface of a polygon with rectangles. Rectangles have to have a specific width, a rectangle can't overlap an other one and each one has to fit in the ...

**7**

votes

**1**answer

628 views

### Three homothetic centers are collinear

I am looking a proof for the problem as follows:
Let a convex hexagon, such that its principal diagonals are concurrent. For each side of the hexagon, extend the adjacent sides to their ...

**10**

votes

**0**answers

306 views

### Determining convexity of a polygon from its Fourier coefficients

Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...

**2**

votes

**1**answer

97 views

### Constructing a polygon of $n$ facets from a set of positive values representing the length of the facets [closed]

The input of my problem is a set of positive values $a=\{a_1,...,a_n\}$ where $n\geq 3$.
I want to construct an $n$-gon where the lengths of the $n$ facets are the values $a_i$ for $i=1,...,n$.
My ...

**2**

votes

**1**answer

175 views

### Area of an irregular, n-sided, non-intersecting (edges) polygon algorithm

I need to generate an irregular, n-sided polygon of non-intersecting edges (n= 200, for example) with the smallest area possible. The position of the vertex is random and I've tried designing a couple ...

**0**

votes

**0**answers

302 views

### 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 ...

**12**

votes

**1**answer

625 views

### 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 ...

**-7**

votes

**1**answer

1k views

### 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 $k$ ...

**3**

votes

**4**answers

467 views

### 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 ...

**0**

votes

**1**answer

343 views

### 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 ...

**4**

votes

**2**answers

431 views

### 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 ...

**4**

votes

**2**answers

303 views

### 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 ...

**6**

votes

**1**answer

213 views

### 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 ...

**10**

votes

**3**answers

737 views

### 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 Howard Masur proved in the 1980's that every rational polygon
(vertex angles rational ...

**5**

votes

**1**answer

1k views

### 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 ...

**2**

votes

**1**answer

142 views

### 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 ...

**2**

votes

**1**answer

146 views

### 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 ...

**17**

votes

**1**answer

457 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 ...

**8**

votes

**0**answers

131 views

### 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 $P$, ...

**0**

votes

**2**answers

1k views

### 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 $\...

**1**

vote

**0**answers

101 views

### 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 ...

**1**

vote

**0**answers

60 views

### 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 length,...

**37**

votes

**4**answers

2k views

### 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 ...

**4**

votes

**1**answer

253 views

### 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 ...

**3**

votes

**0**answers

51 views

### 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 ...

**6**

votes

**1**answer

3k views

### 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 ...