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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2answers
44 views

O.I.M. polygon inequality

I am trying to prove an inequality which was used to prepare the Romanian O.I.M. team. I seem to lack ideas on how to tackle this problem. We take a convex polygon $P_1\ldots P_{n+2}$ and consider ...
1
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2answers
15 views

Geometrical calculation to enlarge the height of rotated rectangle

There is a polygon (rotated rectangle) that defined by 4 corner points in 2D coordinate system. Does anyone help me with the fast (minimum trigonometry operations) algorithm to change its height by ...
1
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2answers
54 views

Area of regular n-gon without trig?

As the title suggests I'm trying to find a formula for the area of a regular n-gon that doesn't use trigonometry. I already know the trig formula and I realize that my question is simply asking for ...
0
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2answers
30 views

Polygon and line intersection

Does anyone help me with the fast algorithm to determine the intersection of a polygon (rotated rectangle) and a line (definite by 2 points)? The only true/false result is needed.
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3answers
40 views

Calculate pentagon area based on lengths of all its sides

Sorry for this question. I guessed there is an online calculator to calculate the area of the pentagon if we know lengths of all its five sides. So, here are the lengths of sides of pentagon ABCDE: ...
3
votes
1answer
18 views

Covering polygons with circles of minimal radius

I have a closed polygon and I would like to fully cover it with a set of K circles of different radius such that the area covered by the circles but outside the polygon is minimal. This seems the ...
4
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2answers
40 views

Area of a polygon inscribed into an ellipse

I have recently found a paper describing that the percentage area error of a polygon inscribed within a circle can be calculated using the following formula. The output of the algorithm is a set ...
1
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1answer
39 views

Finding center of convex polygon

If I'm given vertices of a convex polygon (in the attached image, they are D,E,F,G and H) if we know that inside the polygon there exists a point (say O) for which each angle created by any two ...
2
votes
2answers
60 views

Trying to understand the limit of regular polygons: circle vs apeirogon (vs infinigon?)

In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon: "In the limit, a sequence of regular polygons with an increasing number of sides becomes ...
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2answers
48 views

Calculate area of “hand-drawn” polygon

I have a series of coordinates that represent a hand-drawn polygon. At the intersection, the lines slightly "overshoot," e.g.: ...
2
votes
1answer
73 views

inscribed circle in $n$-gon

If I'm given a circle with radius $r$ and I want to create a polygon with side $n$ (say $n=5$) which can cover the circle fully, then how to prove that a regular polygon is the solution with minimum ...
1
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0answers
12 views

Formula to determine the number of exterior edges in multiple tiled hexagons

I'm looking for a formula which determines the number of external (that is, non-touching) edges in multiple tiled hexagons. By observation, there are 10 external edges when 2 hexagons are adjacent. ...
0
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0answers
14 views

Polygons - necessity of checking for collinearity with edge incident to diagonal's vertices?

I'm reading a book on Computational Geometry ('CG in C' by Joseph O'Rourke). It is quite enlightening but there is one thing I feel like I have to ask about when it comes to triangulation of a ...
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1answer
35 views

Equivalent definitions of symmetry group of regular n-gon (dihedral group)

Let $P_n$ be a fixed regular convex $n$-gon in the plane. For a metric space $M$ we denote by $\text{Isom}(M)$ the set of distance-preserving maps $M \to M$. How can I show that $$ D_n := \left\{\, f ...
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2answers
38 views

Is circle the only Jordan curve with this property?

When I was thinking about one problem that has to do with Jordan curves the problem which I am going to describe now, arose in my mind. And here it goes. It is known that for every $n\geq3$ the ...
1
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0answers
33 views

Solve for Sides of a 5-Sided Irregular Polygon

I have a 5-sided irregular polygon and I know the lengths of 4 of its 5 sides and 2 of its 5 angles. Is there a way to know the length of the 5th side using this information?
0
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1answer
17 views

“How to sort vertices of a polygon in counter clockwise order?”: Computing Angle?

my question relates to the answer to the following question: How to sort vertices of a polygon in counter clockwise order? I don't have a strong background in linear algebra... I don't understand ...
3
votes
1answer
65 views

Nested Regular Polygons

I've created a problem that I do not know how to answer without a huge amount of effort. If you have an elegant solution to it, please share! Take an equilateral triangle of side length 1 and ...
1
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0answers
27 views

Testing if a number N is prime by its regular polygon's angles

Is it possible to tell if a number N is prime by looking at the angles of a regular N-sided polygon? For example, a regular triangle has 60 degree angles, is there a way to tell that the number 3 is ...
1
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1answer
83 views

Intersection of a convex polygon and a moving circle

I have a straight line which intersects a convex polygon in 2D plane. There exists a circle with constant radius. The center of circle is moving on this line. So at first the polygon and circle don't ...
1
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1answer
21 views

Sum of Distances between Points on a Regular $n$-gon

I received a question asking to determine a formula to sum the distances between all points of a regular $n$-gon inscribed in a circle of radius $1$. To solve this, I instead worked with the ...
3
votes
0answers
36 views

Shortest system of roads between 4 cities

You have $4$ cities placed on the vertices of a square of side length $1$ km. You have to come up with a system of roads such that you can reach any city from another (directly or through another ...
1
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0answers
13 views

How to determine the outward normal vector of a face of a hexahedron if the orientation (CW or CCW) of vertices of the face is unknown

Consider a single quadrilaterally-faced hexahedron. If given the co-ordinates of the vertices, $\mathbf{v}_i$, of a face in counter-clockwise orientation, I can compute the corresponding unit outward ...
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1answer
8 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 ...
5
votes
2answers
67 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
53 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 ...
0
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1answer
35 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 ...
18
votes
7answers
644 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
27 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 ...
0
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0answers
32 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 ...
2
votes
2answers
49 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 ...
0
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1answer
28 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 ...
0
votes
1answer
16 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 ...
5
votes
1answer
111 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 ...
0
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1answer
38 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 ...
2
votes
2answers
38 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,
1
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1answer
46 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 ...
0
votes
1answer
32 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
62 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
38 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: ...
2
votes
1answer
58 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
34 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
votes
1answer
26 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 ...
2
votes
1answer
81 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 = ...
0
votes
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?
0
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0answers
52 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
29 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 ...
0
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1answer
46 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
votes
1answer
102 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 ...
1
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0answers
45 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. ...