0
votes
1answer
21 views

Intersections in polygons

I'm having troubles solving the following problem which is about combinatorics : let n a natural number >= 3, and a convex polygon with n vertices. Each vertices are supposed to connect each other ...
6
votes
0answers
40 views
+50

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
0
votes
1answer
20 views

How to sort vertices of a polygon in counter clockwise order?

How to sort vertices of a polygon in counter clockwise order? I want to create a function (algorithm) which compares two vectors $\vec v$ and $\vec u$ which are vertices in a polygon. It should ...
0
votes
0answers
11 views

Permutation of conjoined faces in regular polygon with diagonals

I've been doing some study on relationships in polygons, right now, regular polygons. I've been trying to find relationships between the diagonals, angles, faces, vertices, and primarily conjoined ...
0
votes
1answer
40 views

Parallelograms in a convex hexagon

If you have a (convex) hexagon, and label it ABCDEF, and if ACDF is a parallelogram, and ABDE is a parallelogram, prove that BCEF is a parallelogram also.
1
vote
2answers
191 views

How to find the number of squares formed by given lattice points?

Let us say that we are N integer coordinates (x, y) - what would our approach be if we were supposed to find the number of squares we could make from those given n points? Additionally, if we were to ...
0
votes
1answer
11 views

A polygon compactness metric that filters out the noise of small concavities?

I am trying to characterize the compactness of polygons. I have come across this definition of "compactness ratio" (perimeter/area), which works for most polygons, but I find as the negative concave ...
1
vote
0answers
37 views

Triangulations of the concave polygon

It is known that the amount of possible triangulations of the convex polygon by disjoint diagonals is the Catalan number. But can we somehow know possible amount of the triangulations of the concave ...
0
votes
0answers
63 views

Is there a 4 pointed star that is regular?

I am studying about the area of a 4 pointed star, I wonder if there is really a 4 pointed star that is regular? what could be the characteristics of a regular 4 pointed star? that is how i ...
0
votes
1answer
250 views

Convex n- sided polygon proof writing (homework question)

Would anyone be able to help me with the following problem or give me a push in the right direction? I am not entirely sure where to start and I have been looking at this problem for hours... Any help ...
6
votes
4answers
197 views

Pentagon Geometry

$ABCDG$ is a pentagon, such that $\overline{AB} \parallel \overline{GD}$ and $\overline{AB}=2\overline{GD}$. Also, $\overline{AG} \parallel \overline{BC}$ and $\overline{AG}=3\overline{BC}$. ...
0
votes
0answers
43 views

Can a polygon with minimal perimeter self-intersect?

Recipe. Do the following. Throw $N$ random points $(x_0,y_0),(x_1,y_1),x_2,y_2),\cdots,(x_{N-1},y_{N-1})$ in the plane.Define $(x_N,y_N)=(x_0,y_0)$ : enumeration is $\mod N$ . These points are joined ...
0
votes
1answer
29 views

Selecting a random orthogonal polygon

For a certain demo application, I want to create at random a rectilinear polygon with a given number of corners. Selecting random $x$ and $y$ coordinates of each corner is not a good method, since ...
0
votes
1answer
25 views

Area Of Polygon Whose Edges Are In Given Distance From A Given Polygon Edges

I'm handling a problem which I find quite difficult to solve; My input is a changing number of coordinates (real GPS coordinates), usually I get 4-8 coordinates, and another number,which indicates a ...
2
votes
0answers
42 views

Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...
0
votes
0answers
13 views

For an app teaching about polyhedra, what are some core characteristics to include?

For fun: I'm building a 3d app that teaches about polyhedra. What should I include? The obvious didactic elements for each polyhedron would be: Fundamental polygon's Vertices 
Edges
 Faces
 (and ...
1
vote
3answers
32 views

Find the measure of angle E.

http://static.k12.com/eli/bb/811/7537/0/2_36640_44211/7537/cfcbab7622b25115e3996826ebe54350776a6601/media/a0fb44a9ac3761c0d89bd1c3ffa513c508eb78bf/mediaasset_650483_1.gif help please i still mix the ...
0
votes
0answers
31 views

Volume of a rotated regular polygon

I want to calculate the volume of the shape which is created when you rotate a regular $n$-sided polygon around the $y$ axis with a major radius $r$. (like a torus, but with a polygon as rotated ...
1
vote
1answer
39 views

splitting trapezoid

I have a trapezoid. I know it's height and bases. How can I split it in 2 parts of given area by line parallel to bases. For example my trapezoid has area of S. And I want to get 2 trapezoids with ...
1
vote
1answer
32 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 ...
1
vote
4answers
64 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 ...
3
votes
1answer
31 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 ...
3
votes
2answers
36 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 ...
0
votes
1answer
21 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 ...
0
votes
2answers
54 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 ...
1
vote
1answer
41 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 ...
0
votes
3answers
54 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 ...
2
votes
2answers
50 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 ...
14
votes
3answers
268 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 ...
0
votes
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 ...
1
vote
1answer
38 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 ...
2
votes
2answers
62 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 ...
1
vote
1answer
25 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 ...
1
vote
0answers
26 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. ...
2
votes
2answers
63 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 ...
0
votes
4answers
76 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 ...
0
votes
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 ...
2
votes
2answers
147 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 ...
0
votes
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?
1
vote
1answer
24 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 ...
0
votes
1answer
31 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 ...
1
vote
2answers
55 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: ...
2
votes
2answers
81 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 ...
0
votes
1answer
145 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 ...
0
votes
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. ...
1
vote
1answer
46 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 ...
1
vote
1answer
34 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 ...
0
votes
1answer
94 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 $B, M$ ...
2
votes
1answer
52 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 ...
0
votes
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 ...