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
votes
1answer
54 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 ...
7
votes
3answers
162 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 ...
1
vote
1answer
28 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 ...
0
votes
7answers
159 views

there is any relation between $\pi$, $\sqrt{2}$ or a generic polygon?

I'm a programmer, I'm always looking for new formulas and new way of computing things, to satisfy my curiosity I would like to know if there are any formulas, or I should say equalities, that make use ...
1
vote
0answers
37 views

Relation between polygons

I try to solve a system but I need another equation to make it solved... So I try to find a relation that give the $\beta''$ function of $\beta'$, $\alpha'$, $N$, $D$ et $L$ as described in the ...
1
vote
0answers
11 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
votes
0answers
10 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 ...
18
votes
7answers
634 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 ...
1
vote
1answer
355 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
1
vote
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
vote
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
votes
1answer
14 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
61 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
vote
0answers
25 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
vote
1answer
69 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 ...
-2
votes
0answers
395 views

Number of polygons lying strictly inside a given polygon

There are N simple polygons(non self intersecting polygons) in which no two of them intersect with each other. For any two polygons $P_1$, $P_2$ either $P_1$ lies inside $P_2$ or vice-versa. The ...
1
vote
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
32 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
vote
0answers
10 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 ...
1
vote
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 ...
-1
votes
2answers
43 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
votes
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 ...
5
votes
2answers
66 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 ...
1
vote
0answers
25 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
votes
0answers
30 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
43 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
votes
1answer
27 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
13 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 ...
4
votes
1answer
107 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 ...
2
votes
2answers
35 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,
0
votes
1answer
29 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 ...
1
vote
0answers
35 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: ...
1
vote
1answer
43 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
26 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 ...
1
vote
2answers
60 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.
2
votes
1answer
47 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 ...
1
vote
0answers
29 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 ...
3
votes
4answers
533 views

Can anyone give me x,y coordinates for an octagon?

I am looking to draw a octagon and I need $(x, y)$ coordinates.
0
votes
1answer
23 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
70 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
0answers
51 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 ...
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?
1
vote
2answers
87 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 ...
1
vote
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 ...
8
votes
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 ...
3
votes
2answers
980 views

Prove using integration that $polygon → circle\space \text{as}\space number\space of\space sides → infinity$

Say we have a regular polygon $s$, with number of sides $n$: Is there a way to prove that as $n → ∞,\space $then $s → circle$ using integration?
0
votes
1answer
44 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
68 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
vote
0answers
44 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
votes
1answer
49 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 ...