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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6 views

Adjust Angle to Add Vector for Non-Equiangular Non-Equilateral

I asked this question: Adjust Angle to Add Vector and the solution showed that for equiangular, equilateral triangles the ratio between $\theta$ and $\phi$ was $\pi + \theta = 2\phi$: But now I ...
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2answers
54 views

Can every (convex) polygon be described by a single inequality (involving absolute values)?

For example, $$ |x| + |2x + y| + |x + 2y| + |y| + |x+y| < 4 $$ describes an octagon. I'm wondering whether an equation of this form always exist for any convex polygon, and if so, whether there ...
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2answers
29 views

Construct a regular hexagon of specific height?

Is it possible to construct a hexagon of particular height, meaning distance between the faces (not vertices)? I have seen various methods of constructing a hexagon (ruler and compass only) which are ...
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1answer
34 views

Adjust Angle to Add Vector

Given: Three 2 component vector $\vec{x}$, $\vec{y}$, and $\vec{z}$ such that $\vec{x} + \vec{y} = \vec{z}$ and $\|\vec{x}\| = \|\vec{y}\|$ $\theta$ such that the angle between $\vec{x}$ and ...
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4answers
347 views

Number of ways to connect sets of $k$ dots in a perfect $n$-gon

Let $Q(n,k)$ be the number of ways in which we can connect sets of $k$ vertices (dots), in a given perfect $n$-gon, such that no two lines intersect at the interior of the $n$-gon and no vertice ...
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0answers
21 views

min and max number of hexagons in hexagonal tiling

Is there a way to calculate the maximum and minimum number of hexagons in a hexagonal tiling of a surface with regular identical size hexagons, knowing the area of the surface and the area of the ...
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0answers
17 views

hexagonal tessellation (tiling): uniform distribution of centers of hexagons?

Consider a disk of Radius $R$. We divide the disk into n equal sectors (in the form of pizza slices) . $n= 2^i$ and $i$ is a non-negative integer. Each sector is enclosed with two radii and an arc ...
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2answers
34 views

Total area for a natural nested set of convex polygons.

Suppose we have a convex polygon $P_0$ with $n$ given vertices, and we want to "nest" polygons $P_j$ for $j > 0$ by taking the midpoints between edges of $P_{j-1}$ as the vertices. For a regular ...
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1answer
22 views

What is this octagon constant and how do I calculate it for other 8*N-gons?

I'm drawing a circle with triangles in OpenGL and I am no good at maths. I've tried a couple of ways, one including the simple ...
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1answer
11 views

72-gon with points (cos(k35°), sin(k35°))

This is the question I am given, and I have a model answer for it as well... but I am having difficulty understanding it. What I can see is that the points are on a unit circle. Of course I can ...
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1answer
99 views

Why do rings appear in regular polygons with diagonals?

When looking at regular polygons with all the diagonals filled in, I saw that concentric rings seem to form. Why does this occur? It's not so obvious with small $n$, but for larger $n$ it becomes ...
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1answer
20 views

Distance between the centers of two adjacent hexagons in a hexagonal tessellation

Given a hexagonal tessellation where each hexagon has a inradius r, could we say that the distance between two adiacent hexagons is 2r, and in general the distance between any two hexagons is k2r ...
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2answers
22 views

What is the relation between inradius and circumradius of a hexagon

Let R and r be respectively circumradius and inradius of a hexagon, I would like to know the math relation between R and r. Thanks,
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0answers
25 views

A regular n-sided polygon is inscribed in a circle of radius '1'.If 'An' be the area of that n-sided polygon,find the following.

A Regular n-sided polygon is inscribed within a circle of radius 1, such that the vertices of the polygon touches the circle.If An denotes the area of the 'n-sided polygon',then find the value of A12 ...
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1answer
36 views

Coloring the 6 vertices of a regular hexagon with a limited use per color

I want to solve to following two-part problem. I solved the first part but I am not sure how to start on part B. A) How many ways are there to color the 6 vertices of a regular hexagon using 4 colors ...
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1answer
20 views

Shortest path planning - polygons

Hi there. I am preparing to Robotics class exam. I solved all the questions from previous years exams but I have no clue how to deal with this one. I would appreciate your help very much as no one ...
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2answers
57 views

Why can't we write an equation for a polygon?

You can write an equation for a circle, but why can't you write an equation for a triangle or any other polygon? By equation I mean an equation that is not just a piecewise equation of lines.
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0answers
29 views

Prove special case of Brianchon's theorem using inversion

Brianchon's theorem says: When a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. From interactive demo: ...
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1answer
34 views

Finding vertices of regular polygon

I am trying to find the vertices of a regular polygon using just the number of sides and 2 vertices. After the second vertex, I will make left turns to find each subsequent vertex that follows. For ...
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0answers
21 views

Finding vertices of a hexagon or pentagon

I have a grid of 150000 x 150000 points, and I have a list of points corresponding the x,y coordinates of a shape that make up a slightly imperfect hexagon or pentagon. I'm trying to figure out a more ...
0
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1answer
19 views

Regular polygon Interior angles

I am to find if any given angle(say x)can be interior angle of regular polygon.In other words,is there a regular polygon which angles are equal to X. I know the formula for sum of interior angles of ...
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0answers
10 views

How to clean Self intersecting Polygon ( remove the intersecting points ) [Multipolygon to single polygon]

I have been googling it almost for a week, my problem is that, I have a polygon which is made up of Latitude and Longitude points, I have used Douglus Pecker Algorithm to Decimate the polylines to ...
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1answer
20 views

Check if convex polygon is completely contained completely within another convex polygon.

How can I determine if a convex polygon is completely contained within another convex polygon where speed is critical? I've thought about doing this, which will only use inequalities: pcp = ...
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2answers
28 views

Show that every polygon is limited.

I've already set polygon , polygonal , limited sets . But I have no idea where to start, tried by reductio ad absurdum but did not. Any idea?
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0answers
46 views

On the centroid of a triangle

There's three different ways to see a triangle in the Euclidean plane: as three non-collinear points, say $A$, $B$, $C$; as the line segments connecting the three points, that we can parametrize as a ...
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1answer
22 views

Reference request: Topological space of polygonal chains and its properties

I'm interested in approximations of $C^1([a,b])$-functions by polygonal chains: image File:NURBstatic.svg by User:WulfTheSaxon licensed under GNU Free Documentation License A polygonal chain can be ...
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2answers
654 views

Find the area of this irregular octagon inscribed in a circle [closed]

Find the area of the octagon pictured here I do have some ideas how to solve it, but do not want to write them down here, because I'm hoping to find some different approaches. Also, see 1978 ...
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1answer
42 views

Given the set of all polygons with m sides and perimeter 1, why is there an element with maximal area?

The set of all polygons with $m$ sides and perimeter $1$ has an element with maximal area. I read this fact in a book, and the reference was in German. Does anyone here know? I know how to ...
2
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1answer
46 views

Area of Spherical Polygon

It appears to me that after repeated applications of Girard's theorem on the area of spherical triangles that we can obtain the surface area of a spherical polygon with interior angles ...
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0answers
41 views

Decomposition of ball in Banach Tarski paradox and covering a soccer ball

Banach Tarski paradox says that it's possible to decompose a ball in $R^3$ into a finite number of disjoint subsets, which can be then reassembled into 2 identical copies of the original ball. ...
0
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1answer
35 views

Parallel sides in regular polygons

So I've noticed a couple of things about regular polygons with an even number of sides but I'm having a hard time proving them, these are all very obvious, and I think perhaps induction is the best ...
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1answer
27 views

drawing the polygon

I want to draw the polygon, its sides and area is provided. e.g. number of sides : 4 length of 1st side : 1 length of 2nd side : 2 length of 3rd side : 3 length of 4th side : 4 ...
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28 views

Parameterizing equilateral polygons

I'm not exactly sure how to describe what I want, so if I butcher terms, please forgive me :) I want to "parameterize" the space of simple irregular equilateral polygons with n sides, or at least a ...
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1answer
26 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
40 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
52 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
39 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 ...
3
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0answers
27 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
95 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
23 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 ...
3
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2answers
32 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
20 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
26 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
82 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
110 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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0answers
17 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
34 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
303 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
15 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?