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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12
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312 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
7
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0answers
174 views

What is the shape of the convex $n$-gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $A_n$? ...
4
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26 views

Parameterizing equilateral polygons

I'm not exactly sure how to describe what I want, so if I butcher terms, please forgive me :) I want to "parameterize" the space of simple irregular equilateral polygons with n sides, or at least a ...
4
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0answers
61 views

Polygons with coincident area and perimeter centroids

Let $P$ be a simple, planar polygon. Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the closed shape $P$. Define $c_p$ as the perimeter centroid of $P$, the center of gravity ...
3
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0answers
26 views

Do side-rational triangles of the same area admit side-rational dissections?

Call a polygon side-rational if the lengths of all its sides are rational. Call a dissection of a polygon side-rational if all of the polygons within the dissection are side-rational. Then my ...
3
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26 views

Centroids of a polygon

Obviously, for any polygon we can define at least 3 different centroids: C1: mass center of the lamina; C2: mass center of vertices with equal masses; C3: mass center of the perimeter. For the ...
3
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0answers
80 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 ...
3
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0answers
70 views

Biggest ellipse included in a convex polygon

Considering a N edges convex 2D polygon called P. Let's name its vertices $\{p_1, p_2, ..., p_N\}$ described in a counter-clockwise order, with $p_i = (x_i, y_i)$ What would be, and how would one ...
3
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114 views

Polyonimo Tiling

I came up with the following conjecture the other day, and was wondering if the result was well-known or even true: Define $f(P)$ for a polyomino $P$ (without holes) to be the least number of total ...
3
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0answers
187 views

Formal name for polygon with hole

Is there a formal name for an irregular polygon that has 1 or more holes or cutouts in it? I've heard it refered to as a "swiss cheese polygon" or a "Donut polygon". Is this even strictly a polygon?
2
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33 views

Polygons inscribed in circles, with integer sides and integer radius

Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius? This is what I got so far: All ...
2
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0answers
121 views

Given 3 Vertices of a Tetrahedron, Find the 4th

A regular tetrahedron is circumscribed by the Earth (assume spherical). You are given 3 of the 4 vertices (as latitude and longitude in decimal format), and asked to find the 4th. Any help is most ...
2
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0answers
23 views

The exact type of my 3d model

I have reconstructed vertical features (hole like objects lie on a vertical face) lie on two connected faces. To understand the situation, I say I have 2 walls with many windows and doors on ...
2
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0answers
50 views

How: Determine area painted by a path with width within a polygon

I have a path that represents the movement of some equipment. The equipment has a width so I'd like to determine the approximate area created by this path within a polygon. If I use the distance ...
2
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0answers
46 views

Determine direction of minimum overlap of convex polygons

Given two convex polygons $P$ and $Q$ what is the minimum intersection polygon $A=P\cap Q'$ where $Q'$ is the polygon $Q$ offset by a vector $\overline r$ of fixed length? Put another way, what is ...
2
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0answers
147 views

Area of a Random Polygon

The following is a long description of a computation I'd like to make. You can think of the process described as a spider randomly building a web. I'd like to know how big we can expect the web to ...
2
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0answers
174 views

Points in the cartesian plane

Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed. In the $xy$ plane, suppose $M_{i,j}=\{(x,y)\mid i\le x\le j\}$. ...
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0answers
15 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 ...
1
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0answers
17 views

Hexagonal tessellation: tesselating a pizza slice shape sector of a circle with a special constraint

Consider a sector(in the shape of a pizza slice) of a disk of radius $R$ such that the sector is enclosed by two raddi and and an arc, where the arc subtends an angle $\frac{2\pi}{n}$. We tessellate ...
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0answers
28 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: ...
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0answers
21 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 ...
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0answers
39 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. ...
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0answers
22 views

Number of grid points in a polygon

Following problem: I want to approximate the number of grid points in a polygon, based on the condition that the distance of the grid points are variable. What i need is an approximation, i am aware ...
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0answers
18 views

What is the term to define a single point in a polychoron?

I'm looking for any correct term used to define a point in 4 dimensional space. IE: What does a polychoron compose?
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0answers
77 views

Trigonometry of tetrahedron

I'm trying to develop the algebraic proofs for these two formulas that appear on the webpage below! The image below is of an unfolded non-regular tetrahedron. Triangle B represents the dihedral angle ...
1
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0answers
13 views

create polygon section with equal sides

I have to create essentially these sections of a polygon. I have width(W) and height (H), and number of sides (3 on left abc and 4 on right image ABCD) I need each side to be equal. How can I ...
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0answers
56 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 ...
1
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0answers
62 views

Looking for algorithm for spherical point in polygon that works across meridian and anti-meridian

I need to process millions of latitude/longitude points every day to see if they are located within a defined lat/lon bounded polygon. The polygon may be rectangular, or it may be some irregular 3.. ...
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0answers
16 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 ...
1
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0answers
36 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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0answers
25 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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0answers
47 views

Find all polygons in points in plane

I have a set of points in the plane and I want to find all convex polygons without including a point inside them. For example I want to find all triangles, all four sized polygons, all four five ...
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0answers
44 views

Rectilinear polygons winding around a torus

A simple rectilinear polygon on the plane the difference between the number of interior convex angles ($ 90^{\circ}$) and that of interior concave angles ($ 270^{\circ}$) is always $4$. Consider a ...
1
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0answers
222 views

Question on Proof of Shoelace Formula

I was looking for a way to prove the shoelace formula when I found this proof: For this clockwise order to make sense, you need a point O inside the polygon so that the angles form $OA_{i}A_{i+1}$ ...
1
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0answers
83 views

A convex $n$-gon and the $n$-gon made by its $n$ medians

For a convex $n$-gon $P_1P_2\cdots P_n$, let $M_i$ be the mid-point of the line segment $P_iP_{i+1}\ (i=1,2,\cdots,n)$ where $P_{n+1}=P_1$. Also, let $Q_1Q_2\cdots Q_n$ be an inner $n$-gon made by $n$ ...
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0answers
74 views

Dodecahedron: How do we get the distance between 2 opposite faces?

I am deciphering a CSS code that Ana Tudor Maria has done. http://codepen.io/thebabydino/pen/qIfbL In her example, she has a formula that calculates the distance between 2 opposite faces. I have no ...
1
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0answers
92 views

Area of a 2D convex polytope made of halfspaces

For a computer program I am attempting to solve the area of a convex polytope defined by a finite number of halfspaces. I understand that this forms a polygon and given the vertices of a polygon I am ...
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0answers
37 views

Position on a path

So the situation is: I have a path (which is represented as a two-dimensional array of GPS coordinates) and I have a percentage position on this path. I.e. I know that a person has walked 80% of the ...
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0answers
69 views

Corresponding Triangulations of an (n+2)-gon to n Segments Connecting n+1 Collinear Points

So I'm asked to count the number of ways of connecting n+1 collinear points with n line segments subjected to the following constraints: If the line is L 1) No segment passes below L. 2) Starting at ...
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0answers
133 views

Name and properties of spherical polygon with small-circle sides

Just as the title says: is there a formal name for a convex polygon on a sphere, of which the vertices are connected not by great circle but by small circle segments? My end goal is to intersect two ...
0
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0answers
12 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 ...
0
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0answers
30 views

hexagonal tesselation symmetry

Consider a disk of radius R. We tessellate the surface of the disk using hexagonal tessellation where each hexagon has a circumradius r. The tessellation is made such that the center of one of the ...
0
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0answers
24 views

A regular n-sided polygon is inscribed in a circle of radius '1'.If 'An' be the area of that n-sided polygon,find the following.

A Regular n-sided polygon is inscribed within a circle of radius 1, such that the vertices of the polygon touches the circle.If An denotes the area of the 'n-sided polygon',then find the value of A12 ...
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0answers
9 views

How to clean Self intersecting Polygon ( remove the intersecting points ) [Multipolygon to single polygon]

I have been googling it almost for a week, my problem is that, I have a polygon which is made up of Latitude and Longitude points, I have used Douglus Pecker Algorithm to Decimate the polylines to ...
0
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0answers
45 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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0answers
19 views

Is there an efficient algorithm to extract the farthest ends of a thin contour?

Let's say you have pixel bitmaps that look something like this: From this I can easily extract a contour, which will be a concave polygon defined by a set of 2D points. The question is what is the ...
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0answers
14 views

Polygonal approximation of simple, closed smooth curve.

I was wondering if you could help me with the following problem: Suppose $\gamma : [0, L] \to \mathbb{R}^2$ is an arc-length parametrisation of a simple, closed, smooth curve. If for each $n \in ...
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0answers
15 views

Side length of regular decagon

Ok, so here are the informations. Area: $770 \;\mathrm{cm}^2$ bottom to top: $30.8$ height: half of $30.8$ = $15.4$ Find the perimeter?
0
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0answers
22 views

Do generalizations of the golden/silver ratios have geometric representations in corresponding polygons?

A silver mean of order n is $N_n=\frac{N+\sqrt{N^2+4}}{2}$. For N=1 we get $\phi$, which is found in a regular pentagon. For N=2 we get the silver ratio, which is found in a regular octagon. Will ...
0
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0answers
68 views

How do I detect if two polygons overlap each other or not?

I'm developing a game engine. Currently I'm writing the collision detection part. I have to write down an algorithm which detects if two given polygons are overlapping each other or they are separated ...