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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39 views

What's wrong with this pseudocode for Forster-Overfelt's version of the Greiner-Horman polygon clipping algorithm?

The Problem I'm trying to understand and implement the Forster-Overfelt version of the Greiner-Horman polygon clipping algorithm. I've read the other Stackoverflow post about clarifying this ...
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4answers
93 views

Smallest and largest possible angles of given polygon

What is the smallest and largest possible angle of a triangle? (my guess = 1, 178) What is the smallest and largest possible angle of a quadrilateral (convex or concave doesn't matter, and also ...
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1answer
37 views

finding parallel sides from a irregular decagon?

Is it possible to find out that which of two sides are parallel in this irregular decagon.If,it is yes;then how can I proceed? I have tried with "Consecutive Interior Angles".but can't come to a ...
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0answers
17 views

How do I write my reference for a polygon

Given a regular pentagon ABCDE. Angle EAB = x + y I have this formula: x + y = (1/5)(180)(5-2) But how do I write the ...
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2answers
45 views

What is the ratio of the side length of a regular hepatgon to the side length of the internal heptagon?

Given a regular heptagon with side length 1, create a star heptagon by connecting every vertice. Note that removing the "points" of the star yields a similar heptagon. I want to know the side ...
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1answer
22 views

recutting a polygon to obtain its flip

I am curious if it is possible to cut the triangle with vertices (0,0), (0,1), (1,0) into many polygonal pieces and only translate each piece appropriately to construct the triangle with vertices ...
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2answers
72 views

Derivation of the formula for the area of a regular polygon given the side length.

There are many formulas for finding the area of a regular polygon- but this is the one I am interested in: $$A=\frac{S^2n}{4\tan(\frac{\pi}{n})}$$ where $n$ is the number of sides, and $S$ is the ...
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1answer
43 views

Given a polygon of n-sides, why does the regular one (i.e. all sides equal) enclose the greatest area given a constant perimeter.

This doesn't require much more than the title. I just need an explanation, but an algebraic proof would be a bonus. We can demonstrate this for quadrilaterals, a square is best as shown by this ...
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3answers
63 views

Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ (corrected inequation)

Prove that Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ algebraically or geometrically. $n\sin\frac{2\pi}{n}-n\sin\frac{\pi}{n}$ means the area of a regular n-gon + the area ...
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3answers
55 views

Where is the interior of the polygon?

There is an axis-parallel (orthogonal) simply-connected polygon given as a list of corners. How can I know whether a certain vertical segment has the interior of the polygon on its east or on its ...
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2answers
70 views

Maximum number of vertices in intersection of triangle with box

Suppose we have a triangle and a box in 3D. The intersection of the triangle with the (solid) box will be a polygon with some number of vertices (possibly zero). The vertex count will vary according ...
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3answers
278 views

Area of the Limiting Polygon

Start with an equilateral triangle with unit area. Trisect each of the sides and then cut-off the corners. In this case, we get a regular hexagon - see the picture below. Next, trisect each of the ...
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0answers
14 views

Best convex bounding polygon from a set of given lines

Given a polygon $P$ and a set of predefined lines, I am looking for the subset of lines that creates the best fitting convex polygon with respect to $P$. In other words the area of an ...
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0answers
16 views

Shortest path over polygons with different cost of travel

The real world problem I'm trying to solve: A road has to be built over a terrain. The terrain have areas that are more costly to build over. Those areas can have complex shapes. How do I build the ...
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1answer
36 views

Function of polygonplots does not plot all polygons

Could anyone explain to me why my Maple-Code does not work properly? ...
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1answer
41 views

Area swept by vertex angles in an irregular polygon.

Given any general n-gon, what is the minimum number of vertex angles that sweep the whole area of the polygon? That is to say, what should be the least number of vertices a person has to stand at in ...
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0answers
43 views

Dividing an infinite plane into regions

I am currently working on a computer program for computing layout of graph-based diagrams. Their content is placed in an "infinite" 2D plane with cartesian coordinates in the center of the diagram. ...
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2answers
90 views

Forming a simple polygon from the extrusion of a polygonal chain

Let's say I have a set of vertices connected by edges to form a polygonal chain. Each vertex may be shared by a number of edges to form various sub-chains. An example is shown below. Each edge has ...
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1answer
64 views

Angle between 2 vectors using the determinant

I have a polygon like this: I basically want to find the angles $\alpha$, inside the polygon, between the vectors. I'm using the determinant to calculate the angle alpha: $det(\vec V2, \vec V2 ) ...
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1answer
28 views

Angle between the shortest and biggest diagonals of a Decagon.

I've been doing some geometry lately and approached this problem. I need to get an angle between the biggest and shortest diagonals of a Decagon (10 sided polygon). As the book says I will get only 1 ...
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0answers
28 views

Sufficient condition for a simple polygon

Some background: I'm trying to simulate biological cells as polygons in 2D. Real biological cells have an internal cytoskeleton which enables them to conform to a variety of shapes, many non-convex. ...
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2answers
73 views

How to check does polygon with given sides' length exist?

I have polygon with $n$ angles. Then I have got $n$ values, which mean this polygon's sides' length. I have to check does this polygon exist (means - could be drawn with given sides' length). Is ...
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2answers
49 views

Why must isometry of regular polygon fix origin?

Here is the question: Suppose $\varphi\colon\Bbb R^2\to\Bbb R^2$ is an isometry and $\varphi(\pi_n)=\pi_n$, where $\pi_n$ is the regular $n$-gon with center at origin. Why must $\varphi$ fix the ...
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2answers
121 views

Can a polygon have four 90 degree corners and still not be a rectangle?

On another woodworking forum, someone said that after building a case, you should measure the diagonals to ensure the case is square and that just checking if all the corners are 90 degrees won't ...
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4answers
79 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 ...
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1answer
67 views

Find all points with whole- number's coordinates inside the area of polygon

I've got the polygon with n angles. I know the coordinates of its apexes (their coordinates are integers), but I don't know the total area of that polygon. Is there any way to count how many points ...
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1answer
1k views

Number of Parallel/Not Parallel Diagonals of a Regular Polygon

This is a painfully easy problem, yet the answer continues to escape me. I am seeking a general formula that can be employed to determine the number of diagonals of a regular polygon that are parallel ...
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1answer
13 views

Correctness of the size of an planar integer lattice unknot

A planar integer lattice unknot is a polygon drawn over a two dimensional integer lattice. Here is an example: Given a number $N$, a planar unknot is not always possible. For example, a planar ...
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0answers
24 views

Completeness of moves for polygonal knots

I am going through the paper, MINIMAL KNOTTING NUMBERS, by MANN et. al. On page six of the paper, they defined following moves for polygonal knots. Parallel moves Triangular moves I understand ...
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2answers
165 views

What is inside and outside of complex polygon?

I am reading this paper http://arxiv.org/pdf/1207.3502.pdf Given a complex polygon. Its edges may intersect. The algorithm finds out if given point is inside of polygon or not. It draws a line from ...
3
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1answer
52 views

Mean curvature flow - implementation fails for some meshes

I am working on piece of software to deal with 3D meshes and I need to smooth some meshes. I have implemented MCF by using this formula $\vec{H} = {{t}\over{2}} \sum_{q \in\ link\ p} \vec{Ne} |e| ...
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1answer
74 views

Labeling the vertices of a polygon with 0's and 1's

Suppose $P_n$ is the regular polygon with n vertices ($n\geq 5$). Let $V=\{v_1,\ldots,v_n\}$ be the vertex set. I would like to define a labeling function $\ell:V\to \{0,1\}$ so that ...
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1answer
40 views

Matrices in the plane,polygon assignment, help. please?

a.) Given polygon P with vertices (1,5), (4,8), (8,5), (6,2) and (2,1), find the following: Find the area of P. Tip: Make sure to move COUNTERCLOCKWISE from point to point to ensure you get a ...
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2answers
29 views

Some help needed with a geometry question

What is a formula for all integers n for which a regular polygon with n sides can be constructed using a ruler and compass construction?
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1answer
26 views

Centroid of contiguous polygons

Say that I know which are the centroids of two polygons. These polygons share a number of edges (they belong to a planar subdivision). I want to compute the union of the two polygons and also to know ...
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1answer
32 views

Find the normal of a polygon with vertices that are not linearly independent in 3d

For example, take the vectors: $(1,2,3) (4,5,6) (7,8,9) (10,11,12) (13,14,15)$ What would the normal to the polygon be? I'm guessing it would be $(0,0,0)$? For vertices that are linearly ...
3
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1answer
70 views

How do I create a Hilbert curve that is bounded by a polygon?

All images of the Hilbert curve that I've seen show the Hilbert curve as bounded by the unit square: However, if I have a list of vertices that define a closed polygon, how can I create a Hilbert ...
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2answers
60 views

Relation among the diagonals of a regular heptagon

The question is about this problem (it is from a Math' Olympiad in Germany): Prove that if a regular heptagon $ABCDEFG$ has side 1, then $$\frac1{AC}+\frac1{AD}=1$$ I have found something: ...
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2answers
82 views

Find the area of quadrilateral formed by $4$ (not consecutive) vertices of a $12$-gon inscribed in a circle.

A regular $12$ sided polygon is inscribed in a circle of radius $10$. $A,B,C,D,E$ are its consecutive vertices taken in that order. Find the area of quad. $ABDE$. The angle of $12$-gon is ...
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1answer
184 views

Finding the counter-clockwise direction of points in 3d

I have a set of 5 points of a polygon in 3d. I want to order these points in a counter-clockwise direction. How do I do this? In 2d, to check if two points are ordered counter-clockwise or ...
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0answers
20 views

Regular polygon with $n$ sides , the number of triangles [duplicate]

For a regular polygon with $n$ sides $(n>5)$, the number of triangles whose vertices are joining non-adjacent vertices of the polygon is $n(n-4)(n-5)$. When I take $n=6$, I get David's Star: ...
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2answers
22 views

Help TOm to find the number of black, white and green pieces?

TOm has a spongy boom ball that is made of 32 pieces of polygon figures: 12 black pentagons and 20 white hexagons. Each pentagon adjoins 5 hexagons and each hexagon adjoins 3 pentagons and 3 hexagons. ...
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1answer
66 views

Sides of a quadrilateral

In a triangle, with sides say $a,b,c$ we know that $a+b\geq{c}$ and $|a-b|\leq{c}$. What are the inequalities we can form given the sides of the quadrilateral say $a,b,c,d$ where these are unknown to ...
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1answer
36 views

How does one solve arbitrary polygons, in the same sense as one solves a triangle?

Let us say you are given a polygon, and also are given some, but not all, of its angle measures and side lengths. How would one compute the following: If there is a finite number (zero inclusive) of ...
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1answer
103 views

Forming Combinations of Quadrilaterals From Heptagon

What is the formula for finding Number of Quadrilaterals from heptagon or any other regular polygon above pentagon for that matter?
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1answer
99 views

How to find the circumcircle radius from this following regular hexagon?

Given a regular hexagon $ABCDEF$. We draw diagonals $AC$ and $CE$. Then, we choose two random points inside the hexagon, call that $M$ and $N$, such that: $\frac{AM}{AC} =\frac{CN}{CE}$. If $B, M$ ...
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1answer
54 views

Area of the intersection of regular pentagons

I was playing around in Geogebra and came up with an interesting problem. Take an arbitrary point $A$ and draw a circle with center $A$. Then draw any line through $A$. Call the points where the ...
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0answers
32 views

Is there a way to compute the empty area between a group of touching polygons?

Given a bunch of convex polygons layed out like a house truss, is there a way to compute the empty area, or get a polygon for each of those "holes" between the polygons? I tried starting from any ...
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2answers
1k views

How to calculate perimeter of Polygon with missing the length of one side?

I have following sides(PQRST) of a Polygon where PQ=13, QR=22, RS=8, ST=?, PT= 10 ... i need to find out ST? i don't have any angle i just have the shape? And for calculating perimeter i need to find ...
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1answer
54 views

Equation of a polygon

I need a parametric equation for a filled polygon defined by 3 or more points. The closest I've got is by using 3 points in this equation - $polygon = p1 + u(p2-p1) + v(p3-p1)$. But by using points ...