Questions tagged [polygons]

For questions on polygons, a flat shape consisting of straight lines that are joined to form a closed chain or circuit

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2
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1answer
28 views

segments of a parabolic umbrella

I would like to build a kind of parabolic umbrella. Therefore I am trying to calculate the shape of the fabric parts. My mathematical approach to this problem was to slice a parabolic surface along it'...
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1answer
22 views

Only 5 platonic hedrons, connectivity only proof?

I am reading a course on discrete differential geometry and found this neat problem: After thinking about it for 15 minutes curiosity got the better of me and I cheated. And here's one possible proof:...
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2answers
26 views

Given an n-sided polygon, how would you random sample points within it?

I would like to random sample points within an n-sided polygon. One idea I've thought of is to discretize the area of the n-sided polygon with triangles. I can assign each triangle a unique number in ...
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1answer
19 views

Finding perimeter of polygon inscribed in circle

n equally spaced points are taken on the circumference of a circle of radius 1. How do you find the perimeter of the resulting regular polygon obtained by joining the n points in order?
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23 views

Line picking in a regular polygon

What is the average distance between two random points in a regular polygon with $n$ sides? The length of each side is $l$. (Integration is not allowed). We know: If $n = 4$ (square), the answer is (...
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Raising Polygon Matrices to powers

Suppose we have a matrix $M$ that is of dimension $n * n$. Clearly $M$ has the shape of a square. Considering $M$ can be raised to any $k$-th power, and thus $M^k$ is also a $n * n$ square matrix, is ...
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79 views

Least triangular convex polygon

(This question is based on a question posed in a math riddle post on Reddit.) Let $P$ be a convex polygon. Let the non-triangularity of $P$ be the minimum area of the symmetric difference (shown with ...
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1answer
60 views

Convex set that cannot be approximated by a polygon

In $\mathbb{R}^2$, show that there exist no polygon containing the set $C = \{ (x,y) \in \mathbb{R}^2 | y \geq x^2\}$ and included in $C + B(0,1)$ where $B(0,1)$ is the open unit ball. Intuitively, we ...
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22 views

Splitting a convex polygon of n sides into an odd or even number of smaller polygons.

There is a n sided convex polygon.the number of ways in which we can split this polygon into an a)odd number of smaller polygons such that every vertex of the smaller polygon(smaller polygon means ...
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2answers
71 views

Binomial identity of alternating sum of products of binomial coefficients taken two at a time

I came across this identity a while back and wasn't able to prove it. $$\sum_{i=1}^{n-3}\frac{\binom{n-3}{i}\binom{n+i-1}{i}}{i+1}\cdot(-1)^{i+1}= \begin{cases} 0& \text{if $n$ is odd,}\\ 2& \...
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Inequality related to sides of a convex polygon [duplicate]

The question is: If $a_1,a_2,...a_n$ are the sides of a convex polygon $A_1A_2...A_n$ then prove the inequality: $$\frac{a_1^2+a_2^2+...+a_{n-1}^2}{a_n^2}\ge\frac{1}{n-1}$$ How do I even start ...
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2answers
134 views

Existence of regular $n$-gon whose vertices are arbitrarily close to integer coordinates

I'm self-studying these days about polytopes and I came with this question. I don't know if it's true or not. Let $\alpha_1$, $\ldots$, $\alpha_n$ angles of convex $n$-gon, $n\not=4$. Prove that for ...
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2answers
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What is the area of a regular polygon inscribed in a circle? [duplicate]

For a circle of radius $r$, if I draw a regular polygon with $n \geq 3$ sides (equilateral triangle, square, pentagon, etc.) contained inside the circle, such that each vertex intersects the circle... ...
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Polygon triangulation of regular $n$-gon consisting only of isosceles triangles

As the title suggests today I solved a geometry problem and when I finished I had this question For which integers $n ≥ 3$ can one find a polygon triangulation of regular $n$-gon consisting only of ...
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is that true: shown in image the difference between the convex polygon

I was searching properties of convex polygon and encountered this image, and I feel its wrong. Web page is https://byjus.com/maths/convex-polygon/ this is Google featured website: check google result ...
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1answer
35 views

If an ant jumps from one vertex to another of a regular decagon, based on a condition, which vertex will he be on after his workout?

QUESTION: Brilli the ant stands on vertex $1$ of a regular decagon.. He starts by hopping $1$ space at a time (from $1$ to $2$, then from $2$ to $3$, and so on). He performs $10$ hops in this way. He ...
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What's the name for this star-like shape formed by joining rectangles to the edges of a (regular) polygon?

Is there a proper name for this shape - these examples are all the same, just with different 'arm' counts. I'd describe the second one as a cross, but they presumably can't all be called crosses? All ...
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1answer
25 views

Getting corner coordinates of polygon on a grid

I have a problem in which I need to calculate the coordinates of all corners in a given polygon. The information I have available is a list of tiles/squares within that shape. The shape is always made ...
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1answer
28 views

Relations between a convex polygon and an ellipse

Let $F$ be bounded convex polygon on plane. How can we justify that there exists circle $K$ s.t. if an ellipse contains $F$ and is not contained in $K$, then field of this ellipse is bigger than $1$? ...
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14 views

How many independent variables do we need to construct a polygon

For example, for a triangles we need 3 independent variables to fix the shape (3 sides, 2 sides & 1 angle etc). For a quadrilateral we need 5. I was wondering, if there is a general rule for n-...
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1answer
61 views

Vertices of a regular polygon in the plane with irrational coordinates

Today I was trying to solve a problem about triangulation and I came with this For which $n≥3$ is it possible to draw a regular $n$-gon in the plane ($\mathbb{R}^2$) such that all vertices have ...
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2answers
114 views

The ratio of the area of two regular polygons

The polygons in the figure below are all regular polygons(regular heptagon), share a vertex and the orange line crosses the three vertices of the two regular polygons, the area of the small regular ...
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1answer
37 views

A geometry problem about areas and lengths. Squares and polygons.

An area inequality: Let $ ABCD $ be a square, $ AB = 1 $. Let's consider a 4-sided polygon $ EFGH $, with a peak on each side of the square. The area of EFGH equals $ \frac{1}{2} $. Prove that it ...
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0answers
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Hausdorff distance to alined convex polygon

Consider two such polygons $P_1(x_{1},y_{1},r_1)$ and $P_2(x_{2},y_{2},r_2)$, where $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are the coordinates of the center to $P_1$ and $P_2$, respectively, and $r_1$ ...
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30 views

How to constrain a rectangle within an arbitrary 2d polgyon?

I am working on a graphics project wherein I need to keep a rectangle (assume it has a fixed but arbitrary size, rectWidth * rectHeight) constrained inside an arbitrary polygon. The polygon is ...
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1answer
24 views

Why does the Minkowski Sum of 2 polygons equals to the convex hull of their point's Minkowski Sum? [closed]

I have a hard time solving the question below, I would like to get help or guidance for a solution. The question is at an academic level and deals with computational geometry, polygons and Minkowski ...
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27 views

Generalization of a property of scalene trapezoids

Consider a scalene trapezoid, which is a special case of simple (no crossing boundaries) quadrilaterals (since it has two parallel sides). When one removes a vertex from this trapezoid and reconnects ...
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1answer
45 views

Solve degrees of freedom of prototile angles for a given pentagonal tiling

I am trying to find exact instances of a pentagonal tiling, but I am struggling to solve all the angles for a given type. Taking type 8 as an example: where: $a = b = c = d$ $C = 360 - 2B$ $E = 180 -...
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1answer
55 views

Circles and enneagon

Using that in a triangle ABC, $\tan\frac A2=\frac{r}{p-a}$ where $p=\frac{a+b+c}{2}$, I found that the radius are equal if $\tan^220°=\frac{\frac{1+2\cos40°}{\cos20°-1}}{1-\frac{1}{2\cos40°}+\frac{\...
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1answer
49 views

Prove that the angles of exactly $4$ vertices of a polygon, with prime number of sides, can form an arithmetic progression

QUESTION: Let $n \ge 4$ be a natural number. Let $A_1A_2 \cdots A_n$ be a regular polygon and $X = \{ 1,2,3,\cdots,n \} $. A subset $\{ i_1, i_2,\cdots, i_k \} $ of $X$, with $k \ge 3$ and $i_1 < ...
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1answer
29 views

Formula for polygon dimension

In this example, if all sides are 10mm long and I did not know that the distance between the parallel lines was 17.321 what formula could be used to get that number?
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1answer
20 views

Fixed area polygon maximize the area of its convex hull

For a fixed area, what kind of polygon that would obtain the maximum area of its convex hull. For e.g. area 1 square, since it’s already convex, so the convex hull having area 1 too. Would there be ...
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1answer
54 views

Polygon Diagonal Combinatorics

A diagonal for a polygon is defined as the line segment joining two non-adjacent points. Given an n-sided polygon, how many different diagonals can be drawn for this polygon? I know that the number of ...
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19 views

About polygons, diagonals and areas

Let's call the diagonal of a polygon “good” if it splits it into $2$ parts of equal area. $3$ good diagonals split the convex heptagon into several parts. Prove that there are no pentagons among ...
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1answer
29 views

Having (x,y,t) determine if two persons were near for more than 5 minutes

Having a dataset/timeseries consisting of $x,y$ and time for n people I need to determine whether one person was close to another for more than m minutes. Consider that $x,y$ are Real numbers, so if ...
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0answers
23 views

Are points on the edge of a polygon inside the polygon?

This seems like a simple question but I did not find a clear answer to this by Googleing. For a simple polygon that is not self-intersecting, such as a triangle or square, if an arbitrary point is ...
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1answer
75 views

What do you call a “circle-ish” polygon with 256 sides?

I would like to find an image of such a geometric shape or generate one myself, but I am not sure what to look for.
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3answers
115 views

Does this arrangement of polygons necessarily contain a hole?

Consider an arrangement of finitely-many open polygons in the plane (not necessarily convex) such that each polygon intersects at least two other non-intersecting polygons. Is there always a sub-...
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0answers
34 views

Can the bronze ratio be found in a regular tridecagon?

The golden ratio $\frac{1+\sqrt 5}2$ is found in the regular pentagon with side length 1 as the length of the first diagonal The silver ratio $\frac{2+\sqrt 8}2$ is found in the regular octagon with ...
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1answer
67 views

Side of a square inscribed in regular polygon

I'm trying to derive a general formula for the side of a square inscribed in a regular polygon with $n$ sides. I know there may be more than one, e.g. infinitely-many for an octagon, but I want to be ...
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1answer
25 views

Given a straight pyramid with a regular hexagonal base, we pass through the center of its base an alpha plane parallel to a side face.

Given a straight pyramid with a regular hexagonal base, we pass through the center of its base an alpha plane parallel to a side face. Find the ratio between the area of ​​the obtained section and the ...
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1answer
14 views

The polygon width parallel to the x axis as a function of the y ordonate?

Considering a polygon with n vertices as input. I need to calculate the integral of the form \[\int_A p(y) dA \] where $p(y)$ is a piecewise polynomial function of $y$. May be if I could find the ...
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0answers
26 views

On Erdös-Szekeres convex polygons lower bound

I have problems with the construction of $2^{n-2}$ points that contain no n-gon, particulary, the proof of the book "Open Problems in Mathematics". The proof sais that: For $i = 0, ..., n-2$ ...
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5 views

What is the distance of a polygon to the closest convex regular polygon of fixed circumradius?

In the following I assume polygons to be described by 2d points representing their vertices' positions in the 2d plane. Let $p$ be an $n$-gon and $Q=\{q\}_R$ be the family of regular $n$-gons with (...
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1answer
33 views

Intersection and containment detection for convex polygons

I need to check if two convex polygons intersect of not, in a efficient way. The description of the intersection itself is not required. I know the method of separating axis, but it takes $NM$ tests ...
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0answers
48 views

Construct a regular pentagon

I want to construct a regular pentagon with side length $4$. I have done the following: We are given the segment $|AT|=4$ and we will create a point $B$ outside of $|AT|$ such that $|AB|$ is divided ...
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1answer
40 views

Let $A_1A_2…A_n$ be the regular polygon incribed a circle $(O)$ with radius $R$ what is value $k=\frac{\sum_{i<j} A_iA_j^\alpha}{R^\alpha}$?

Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$ with radius $R$, $\alpha$ is real number, let $$k=\frac{\sum_{i<j} A_iA_j^\alpha}{R^\alpha}$$ My question: I am looking for ...
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2answers
36 views

planar loop shapes

I know if I stick two pins on a paper, and trace a taut loop around them, I get an ellipse. With one pin, I get a circle. Question is, are there names for shapes I get if I trace a taut loop around 3, ...
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0answers
24 views

Algorithm to generate points of a circular polygon with N

I'm programming a game and I want to create polygons that will act as the boundaries for this top-down game. I want to have circular boundaries like a hexagon, pentagon etc. which I can create as long ...
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0answers
9 views

Where do I add supports to a polygonal cut-out?

Given a line which represents a cut into a 2D surface, where would I place supports so that it is stable? For example, say I want to cut a circle out of a piece of cardboard, but not entirely. I ...

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