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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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Find a best fit line for set of points using convex hull algorithm

A line is a best fit for a point set S in the plane if it minimizes the sum of the distances between the points in S and the line. Assuming a convex hull algorithm is available, find the best fit line ...
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42 views

Cut a convex polygon in two equal areas with minimum perimeter

Given a convex polygon. How to find a cut that divides the polygon in two equal area parts and the length of this cut is minimum. Possible solution is 1. Find a minimum polygon projection. 2. Have ...
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3answers
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Single-Line Equation for Equilateral Triangle

Is it possible to come up with a single-line equation in rectangular coordinates for an equilateral triangle with circumradius $R$, positioned symmetrical about the $y$-axis, as shown in the diagram ...
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What does the determinant of 3x3 matrix mean? how does the determinant tell me the orientation of polygon?

I need help with understanding the problem. The coding isn't really the problem rather the math part. I don't understand what a delta test method that tells me determinant of a 3x3 matrix is supposed ...
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27 views

Solid angle created from irregular polygon (over a sphere)

I have an $n$-polygon on a sphere ($n\geqslant3$). In this example the vertices are $C,D,E,F,G,H,I,J,K$. Which solid angle alpha generate this polygon respect origin of the sphere? For $C,D,E,F,G,H,I,...
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38 views

Chord partition of regular polygon: same fraction of area and perimeter?

This is a variation of a question posed by James Tanton on Twitter. Let $P$ be a regular $n$-gon, $n \ge 3$. A chord $c$ of $P$ is a segment connecting two distinct points of the boundary of $P$, on ...
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1answer
59 views

Why is the sum of all external angles in a convex polygon $360^\circ$ and not $720^\circ$?

Why is the sum of all external angles in a convex polygon $360^\circ$? From my understanding, for each vertex in a convex polygon, there exist exactly $2$ exterior angles corresponding to it, which ...
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3answers
37 views

Cyclic pentagon $ABCDE$ has radius three. If $AB=BC=2$ and $CD=DE=4$, find AE. [closed]

Given cyclic pentagon $ABCDE$ with the radius $3$. If $AB=BC=2$ and $CD=DE=4$, find $AE$. I think trigonometry tricks are really useful on this problem, but I still can't get the final answer.
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24 views

Maximum surface area of polygons with sequential side lengths. [closed]

What is the maximum area of a polygon with sides of lengths $1,2,3,\ldots,N$? Intuition tells me the polygon must be inscribed in a regular polygon with $1+2+3+ \cdots +N$ sides. What would be the ...
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1answer
61 views

Algorithm to construct irregular polygon

I have number of line segments (they represent walls in floor scheme) each accompanied with length and adjacent angle. What sequence of steps should my algorithm perform in order to obtain set of ...
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Non-classical examples for generalized quadrangles

I have been told that there is no non-classical example for n={6,8} known yet for quadrangles. Could you share some study on it?
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1answer
46 views

general equation for a n-side regular polygon

I was revisiting some geometry problems, and i got me thinking if there is any king of general equation to describe a n-side polygon? Some way similar to the equation that describe a circle, which we ...
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27 views

Polygon with all sides with different lengths

I searched a lot but I could not figure out the property name of a polygon with all sides with different lengths. First of all I am assuming that it is possible to have at least one configuration for ...
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1answer
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Numerical Analysis - $n$-sided polygon tangential

i need help with this question..I'm not so sure how to go about the arguments. Any help would be appreciated. Consider a regular $n$-sided polygon tangential to and enclosing the unit circle to ...
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23 views

Polygons defined by lengths of sides

Can a polygon be defined by the lengths of its sides? In other words, if given the lengths of the sides of a polygon, is there a way to figure out what polygon has those lengths and show that there ...
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1answer
29 views

Which of two vertices has the wider angle?

Given that: two vertices from different paths share the same point; the winding orientation of these paths is unspecified; the angle at both vertices will be less than 180 degrees these paths don'...
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2answers
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Plotting points on a halfcircle, given diameter and facing direction.

I know the coordinates of point $1$ and $2$ and some radius $r$ at a halfcircle with centerpoint point $1$, with the gap of the halfcircle pointing towards point $2$. How do I compute the (lets say $...
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1answer
53 views

Check if latitude & longitude coordinates are inside a specific range of a latitude longitude polygon

Since I'm not a mathematician I came here to ask what the most efficient way is to check if latitude and longitude coordinates are inside a range (for example 50 meters) of multiple latitude and ...
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1answer
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Internal angles in regular 18-gon

This (seemingly simple) problem is driving me nuts. Find angle $\alpha$ shown in the following regular 18-gon. It was easy to find the angle between pink diagonals ($60^\circ$). And I was able to ...
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1answer
36 views

What is the length of the hypotenuse?

We have $n$ isosceles-right-angled triangles. The hypotenuse of the $n^{\textrm{th}}$ triangle is the base of the $(n+1)^{\textrm{th}}$ triangle. For the first triangle, $T_{1}$, the length of the ...
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1answer
17 views

Calculating square meter area with polygonal geographical coordinates (metric - not DMS system)

I'm working on a program but my problem is not on software side but mathematical. I have the following input : ...
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24 views

Finding the area of a polygon with complex numbers.

If we consider a regular polygon defined by the $n$-tuple $Z=(z_0, z_1, \dots, z_{n-1})$, and set $$A(Z)=\frac{1}{2} \Im \left ( \sum_{k=0}^{n-1} \bar{z_k} z_{k+1} \right )$$ Given that the number $...
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Construction involving regular polygons inside a circle

Let's make a construction involving regular polygons: ► First, we begin with a equilateral triangle, with side $\ell_3 = 1;$ ► After, we draw a square on the middle point each side of the initial ...
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1answer
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Number of right angled triangles formed by vertices of a 14-gon

Here's a question that I found on the website of International Kangaroo Maths Contest. The question goes like this: What is the total number of right angled triangles that can be formed by joining ...
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1answer
29 views

Describe/name a cube in different poses

I have a relatively simple question: are there different names for a cube in different poses? Specifically, I need to distinguish a regular cube (i.e. a cube lying flat in the XY plane and parallel ...
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1answer
22 views

Getting slice number of regular polygon from coordinates

For an n sided polygon divided up into n triangles, i want the triangle "index" from a coordinate inside the polygon Example: The coordinates at the green dot should give 2, the red dot should give ...
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2answers
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Prove that $|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2$. [closed]

A polygon $A_{1}A_{2}...A_{n}$ has a circumscribed circle with radius $R$. Prove $$|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2.$$
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1answer
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Self-Similar Polygon Tessellations

It is well-known that the only regular polygons which tessellate the plane (using only one shape) are the triangle, square, and hexagon. However, there are many more tessellations of the plane by ...
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How to find the largest regular shape inside irregular convex polygons in 2D

I want to find a way that can find a regular or common shape (e.g a circle, triangle, square or rectangle) that has the maximum area inside different irregular 2D polygons. From this article, it ...
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0answers
56 views

Fill a polygon with triangles with certain criteria

I have a random polygon like the example below (in general the polygon can have n vertices where n > 3). I want to find set of 3 vertices that constructs triangles inside the polygon in a way that ...
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How to find a proper bivariate Gaussian distribution that integrals 95% over a convex polygon

I have a known 2-D convex polygon, and my goal is to find a bivariate Gaussian distribution, that 1) samples from the Gaussian falls into the polygon 95% of the times, and 2) should be reasonably ...
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Set of hyperplanes to polyhedron topology

Problem By intersection of a set of planes (in three-dimensional space) I constructed an edge-vertex topology structure (see image). I was able to reconstruct the polygons in this structure by ...
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105 views

Using Lagrange multiplier for n-polygon (Dido's problem)

We have an example of the discrete version of Dido's problem. Let $n\in N$ and $n \geq 3$. A polygon exists with $n$ sides and corners P. The corners are given by: $P_k$ = ($r_k \cos \sigma_k$, $...
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1answer
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How to find coordinates of non intersecting areas between two intersecting polygons?

I have two polygons that have some intersection. Then there are some areas which belong to either Polygon1 or Polygon2. How can I find the coordinates of those areas? Is there any algorithm to do ...
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Calculate area of random polygon with specified 3D coordinates of boundaries

Introduction I am a Belgian software engineer working in a company that is producing press brakes. I now have an interesting problem, where I would like to know the best solution, performance is ...
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Area of irregular similar hexagons [closed]

Irregular Hexagons $A$ and $B$ are geometrically similar. The shortest sides are $4$ inches and $3$ inches, respectively. If the area of hexagon $A$ is $48in^2$, what is the area of hexagon $B$? I ...
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1answer
20 views

Differing notions of 'degrees' in 2D polygons

Suppose a radius sweeps a semihexagon inscribed in a semicircle. Has it swept $180\unicode{xb0}$ (semicircle) or $360\unicode{xb0}$ (semihexagon)?
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2 hexagons that are not affinely isomorphic.

I'm reading a book that provides an example of two hexagons that are not affinely isomorphic. One is a regular hexagon, and the other is not. The book defines two polytopes (in this case hexagons) ...
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Determining the smallest set of measurements that uniquely define a polygon

I am trying to determine the smallest set of measurements that uniquely determine an $n$-sided polygon, symmetrical solutions being counted as one. I was able to find without proof or reference that "...
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0answers
51 views

How can we cut out $B'$ from $A$ such that $B'$ is similar to $B$ and $B'$ is as large as possible?

Given any arbitrary, closed shapes $A$ and $B$ on the Euclidian plane, how can we cut out $B'$ from $A$ such that $B'$ is similar to $B$ and $B'$ is as large as possible? This question, I found it in ...
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1answer
33 views

For a polytope, why is the intersection of faces a face?

First I fix definitions: A polytope in $V$ is a subset that is bounded and is the intersection of half-spaces of $V$. A half-space is defined with respect to an affine form, and is that part of $V$ ...
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2answers
36 views

problem of dissecting a 1 x k rectangle into similar but incongruent polygons

I have this problem from USAMO 2004: "For what values of $k > 0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but incongruent, polygons?" I feel that all values of k will ...
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2answers
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How to get remaining unit normals by purely geometric reasoning instead of calculating?

Here "~" represent the vector. Draw the tetrahedron AOBC through vertices A(1; 0; 0), the origin O(0; 0; 0), B(0; 2; 0) and C(0; 0; 3) in the standard cartesian 3D-frame. Calculate the outward unit ...
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2answers
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Polyhedrons exclusively made out of even sided polygons

I know that the cube is the only 3 d shape which falls in polyhedrons but still is composed of squares, exclusively although its a even sided shape. I have noticed that after square there is no single ...
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2answers
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Circumradius of a regular heptagon

My graphing software says that the value of circumradius of a regular heptagon, of side unity, upto 5 decimal places, is 1.15238. Just as the circumradius of a regular pentagon of length unity can be ...
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1answer
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Where to place 10 guards in a 17-walled art gallery such that removal of any one leaves the gallery unprotected?

I am stuck on a problem related to art galleries. It is problem $12$ in the chapter about art galleries in How to Guard an Art Gallery and Other Discrete Mathematical Adventures by T.S. Michael (I ...
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35 views

Constrain to express points form a convex polygon

Is there a way to characterize convexity of a polygon from the coordinate of its vertices? In other words, is there a relation a set of points, thought as vertices of a 2d polygon, must satisfy to ...
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1answer
24 views

Exclusive disjunction of rectilinear polygons

Polygons all of whose edges meet at right angles are called rectilinear polygons. Furthermore, lets assume integer coordinates at each point. The question is: Can we give an upper bound about the ...
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Change of angle inside a quirky hexagon

So I am dealing with the hexagon as shown in the picture below and I need to find out how one angle depends on another angle. More specifically, I need $\frac{d\psi}{d\varphi}$ at $\varphi=0$. Note ...
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2answers
66 views

Angle chasing with a marked point within a regular hexagon

A point $P$ is marked within regular hexagon $ABCDEF$ so that we have $\angle BAP=\angle DCP = 50^\circ $ If $\angle APB$ has measure $x$ degrees, find $x$. Here's a diagram: Referring to the ...