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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32 views

ratio of “diameter” of a n-gon to perimeter

So say I have a regular polygon with n sides, and I bisect the an angle E such and the line (EF). Assume line segment EF has length b, while the polygons side length is s. What is $b/(n*s)$, and as n ...
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1answer
45 views

What is the name for a rectangular figure of many sides?

What is a polygon where each edge is at a 90 degree or 270 degree angle to the prior edge (giving both concave and convex vertices) called? Here is one example of such a shape: ...
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1answer
75 views

Regular polygons.

$1.)$ What is the minimum measure of exterior angle possible for a regular polygon. $2.)$ What is the maximum measure of interior angle possible for a regular polygon.? $3.)$ And how many sides ...
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1answer
49 views

Polygon center which always lies inside the polygon (with no hole)

Is there is any type of centre (of polygon) which always lies inside the polygon (with no hole)? Note: Here our polygon may be any type of polygon (convex or concave) but ...
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0answers
32 views

Do side-rational triangles of the same area admit side-rational dissections?

Call a polygon side-rational if the lengths of all its sides are rational. Call a dissection of a polygon side-rational if all of the polygons within the dissection are side-rational. Then my ...
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2answers
138 views

How many sides can a polygon have before it is “considered” a circle?

Good day, my family had a dinner discussion about polygons and how many sides a polygon has in relation to the angle measurement you'll get when you measure an "arc" encompassing a "side" of the ...
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0answers
33 views

Number of grid points in a polygon

Following problem: I want to approximate the number of grid points in a polygon, based on the condition that the distance of the grid points are variable. What i need is an approximation, i am aware ...
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2answers
33 views

is a line of symmetry going through the center of an octagon parallel to the ground it is on?

I came across this question recently, It wanted me to find angle EDG assuming the line crossing between the octagon/polygon is it's line of symmetry. My answer was 25 degrees, but A few friends ...
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0answers
21 views

Is there an efficient algorithm to extract the farthest ends of a thin contour?

Let's say you have pixel bitmaps that look something like this: From this I can easily extract a contour, which will be a concave polygon defined by a set of 2D points. The question is what is the ...
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0answers
31 views

Centroids of a polygon

Obviously, for any polygon we can define at least 3 different centroids: C1: mass center of the lamina; C2: mass center of vertices with equal masses; C3: mass center of the perimeter. For the ...
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1answer
90 views

What is a composition of two binary relations geometrically?

the composition was defined as follow: (a,b) \in (R;S) <=> there is c | (a,c) \in R and (c,b) \in S . If our two relations R and S are two convex polygon ...
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2answers
189 views

How to draw an irregular polygon given all the side lengths and total area, but no angles?

I need to draw a sketch of an irregular piece of land where I know the 8 side-lengths and the total area, but I have no information on the interior angles. The description of the terrain is as ...
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24 views

Polygonal approximation of simple, closed smooth curve.

I was wondering if you could help me with the following problem: Suppose $\gamma : [0, L] \to \mathbb{R}^2$ is an arc-length parametrisation of a simple, closed, smooth curve. If for each $n \in ...
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0answers
18 views

What is the term to define a single point in a polychoron?

I'm looking for any correct term used to define a point in 4 dimensional space. IE: What does a polychoron compose?
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1answer
36 views

Why simple polygons in plane have this property?

If we are given a simple polygon $P$ in the plane by the points $A_1, A_2, \dots, A_n$. How can we prove that there are $3$ consecutive points $A_i, A_{i+1}, A_{i+2}$ (if $i = n$, for $A_{i + 1}$ and ...
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1answer
385 views

Drawing ellipse as google.maps.Polygon with 8 points

In a web page using Google Maps JavaScript API v3 (including Geometry library) I currently draw an ellipse as a "diamond" with 4 corner points by the following JavaScript code: ...
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0answers
17 views

Side length of regular decagon

Ok, so here are the informations. Area: $770 \;\mathrm{cm}^2$ bottom to top: $30.8$ height: half of $30.8$ = $15.4$ Find the perimeter?
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1answer
37 views

Regular polygon finding the circumference

Area of the 10 regular polygon is 770cm^2 and the gap between the parallel line is 30,8cm. Find its circumference. Yeah, I'm going to fail on this subject.
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4answers
710 views

What polyhedron has 11 vertices and 17 edges

On my math test it asked me how many polygons it takes to create a polyhedron that has $11$ vertices and $17$ edges. I'd just like to see what the shape would look like and I can figure out the ...
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2answers
51 views

How to connect a line between 4 randomly placed points on a plane such that the line does not cross itself

You get 4 coordinates of points on a plain. You need to connect them all with a line. The line must not cross itself. What's your strategy?
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1answer
9 views

Regular $n$-gons and Linear Tranformations

Let $P$ be a regular $n$-gon and let $Q$ be an another convex polygon with the same number of vertices. Is there always a linear transformation $M\colon \mathbb{R}^2 \to \mathbb{R}^2$ such that $M(P) ...
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1answer
66 views

Discrete Gauß and geodesic curvature

Imagine that you have an n-polygon $S$ and you wanted to calculated the discrete Gaussian or gedoesic curvature. How are they defined? If $p$ is a vertex of $S$ then Gauß-Bonnet suggests that the ...
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0answers
23 views

Do generalizations of the golden/silver ratios have geometric representations in corresponding polygons?

A silver mean of order n is $N_n=\frac{N+\sqrt{N^2+4}}{2}$. For N=1 we get $\phi$, which is found in a regular pentagon. For N=2 we get the silver ratio, which is found in a regular octagon. Will ...
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1answer
45 views

Find a segment inside a polygon

Let $P = A_1 A_2 \dots A_n$ be a polygon in the plane, with $n \geq 4$. That is, it is a closed broken line with no self-intersections. Then $P$ determines an interior and an exterior, by the Jordan ...
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1answer
78 views

Interior angles of a polygon.

I was solving problems from Paul Zeitz's book "The Art and Craft of Problem Solving." There is a problem which states 3.2.11 Fix the proof in Example 2.3.5 on page 45. Show that even a concave ...
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3answers
361 views

Scaling and rotating a square so that it is inscribed in the original square

I have a square with a side length of 100 cm. I then want to rotate a square clockwise by ten degrees so that it is scaled and contained inside the existing square. The image below is what I'm ...
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71 views

Polygons with coincident area and perimeter centroids

Let $P$ be a simple, planar polygon. Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the closed shape $P$. Define $c_p$ as the perimeter centroid of $P$, the center of gravity ...
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2answers
90 views

Number of $r$-sided polygons in $P$ with no common edges

We have a $n$-sided convex polygon $P$. How many $r$-sided polygons $(r<n)$, with its vertices among those of $P$, can be formed such that it has no sides (edges) in common with $P$? I tried ...
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3answers
168 views

Transform polygons into one another?

I am aware that there must be no standard way to achieve this, but I don't know what has been done so far. I feel like I'm missing keywords to investigate further. I have any two 2D polygons $a$ and ...
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1answer
67 views

How to calculate the center of a regular polygon?

What is the formula for the center of an n-edge regular polygon that has the given segment as its edge? So, given a segment AB, ...
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0answers
95 views

How do I detect if two polygons overlap each other or not?

I'm developing a game engine. Currently I'm writing the collision detection part. I have to write down an algorithm which detects if two given polygons are overlapping each other or they are separated ...
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1answer
69 views

Fattest scalene quadrilateral

What angles of a plane scalene quadrilateral maximize its area? By 'scalene' I mean the four lengths are unequal. It is known that if a quadrilateral has opposite sides equal and parallel as a ...
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0answers
32 views

Projection of a Triangle into a Tetrahedron

I was referring to a paper to implement an algorithm in which one of the step was to project the triangle into the ...
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1answer
47 views

Calculating rectangle having 2 coordinates and 1 length

How could I calculate the other 2 points if I have a rectangle with 2 points and 1 length given? ...
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1answer
65 views

Inner angles in polygons

A point of a polygon is called convex if it's inner angle is less than 180 degrees. Prove that in every simple polygon there is at least one convex point.
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43 views

Two convex polygon intersection from set of m convex polygons with total n vertices

I have a set of m convex polygons $(p_1,p_2, \ldots p_m)$. $n_i$ is the number of vertices in $p_i$. $\sum_{i=1}^{m} n_i = n$. Each polygon has vertices listed in anti-clockwise direction, starting ...
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1answer
36 views

Visible objects from a point in a polygon with holes in 2d

The problem is the following: Given a polygon P with h holes/objects and a point c inside P but outside the holes/objects. P has n given vertices and each hole/object h has 4 vertices (the ...
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1answer
38 views

Minkowski sum of convex sets in the plane which are not polygons

Can the Minkowski sum of two convex sets in the plane which are not polygons be a polygon? Explicitly my convex set is of the form $ C= \{(x,y) \in \mathbb{R}^2 : x,y \geq 0 \text{ , } ...
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1answer
164 views

Use of Delaunay Triangulation and Voronoi Diagram to find alpha shape using Edelsbrunner's algorithm

I am learning how to find the shape of a set of points in 2-D. I understand that Alpha Shape method is a good way to find the shape of a set of points. Alpha Shape was originally introduced by H. ...
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1answer
147 views

Can the $9$ point circle be generalized to $n$-gons of $n\gt3$?

All triangles have concyclic vertices and have a $9$ point circle which intersects the triangle's feet and the midpoints of its sides (as well as $3$ other significant points). Is this special for ...
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1answer
51 views

question about a special case of an $n$ sided polygon

Here is an interesting question that I have been thinking about for awhile now but do not know the answer to. Suppose you have a convex polygon with $n$ sides. What would be an example of such a ...
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2answers
398 views

Sorting a list of points in 2-D clockwise

I have number of points with co-ordinate (latitude, longitude) in 2-D: Here is a collection of some points: \begin{array}{ccc} \hline No.& lon & lat \\ \hline 1& 84.07921& 24.49703 ...
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1answer
141 views

What is the number of intersections of diagonals in a convex equilateral polygon?

Question: [See here for definitions]. Consider an arbitrary convex equilateral polygon with $n$-vertexes ($n\geq 4$) and the $n$-sequence $\langle \alpha_i~|~i<n\rangle$ of its angles which ...
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0answers
92 views

Trigonometry of tetrahedron

I'm trying to develop the algebraic proofs for these two formulas that appear on the webpage below! The image below is of an unfolded non-regular tetrahedron. Triangle B represents the dihedral angle ...
3
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1answer
61 views

In every polygon circumscribed about a circle, there exist three sides that can form a triangle.

How can one show that in every polygon circumscribed about a circle, there exist three sides that can form a triangle? (This was posted by another user and then deleted while I was typing my answer.) ...
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1answer
59 views

Beautiful property of every single circunscribed polygon ever

Show that in any circumscribed polygon, there exist three sides which could form a triangle. Been on it for a while starting with quadrilaterals and trying to connect some properties and proved for ...
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0answers
14 views

create polygon section with equal sides

I have to create essentially these sections of a polygon. I have width(W) and height (H), and number of sides (3 on left abc and 4 on right image ABCD) I need each side to be equal. How can I ...
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1answer
38 views

Intersections in polygons

I'm having troubles solving the following problem which is about combinatorics: let $n$ be a natural number $\ge 3$, and a convex polygon with $n$ vertices. Each vertices are supposed to connect ...
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1answer
274 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
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1answer
478 views

How to sort vertices of a polygon in counter clockwise order?

How to sort vertices of a polygon in counter clockwise order? I want to create a function (algorithm) which compares two vectors $\vec v$ and $\vec u$ which are vertices in a polygon. It should ...