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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1answer
63 views

Can one use Pick's theorm to prove that area size 5 covers at least 6 grid points?

According to Pick's Theorem, the size of an area $A$ can be calculated by the sum of the interior lattice points located in the polygon $i$ and the number of lattice points on the boundary placed on ...
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1answer
18 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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1answer
87 views

Furthest point on regular polygon given arbitrary direction

In a circle of radius $r$ centered at $c$, if I want to know the point on the circle that is furthest in a direction specified by a vector $d$ I use the formula $c+(r/||d||)d$. Is there a similar ...
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1answer
145 views

Prove using integration "circle is a polygon when number of sides-> infinity

Is there a proof of "if number of sidesof a regular polygon ->infinitythe regular polygon -> circle." using integration?
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1answer
115 views

Shortest path calculation

I have a given set of start points, a given set of end points. Each start point corresponds to one endpoint. I have to visit all start points, and then the corresponding end points, in the most ...
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1answer
83 views

sum of exterior angles of a closed broken line in space

I am looking for a simple proof of the following fact: The sum of exterior angles of any closed broken line in space is at least $2 \pi$. I believe it equals $2 \pi$ if and only if the closed broken ...
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1answer
76 views

Largest Convex polygon consisting of k points

The problem is Given a set of points, determinate the Largest (in terms of area) Polygon consisting of at most $k$ points. In a shape like The one below: $k = 3,polygon =A,F,G $ I would like to ...
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0answers
256 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 ...
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151 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 $A_n$? ...
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76 views

The Erdős-Szekeres problem on points in convex position

The Erdős-Szekeres problem on points in convex position and its proof using Ramsey theorem are well know. The problem goes like this: For every natural number $k$ there exists a number $n(k)$ such ...
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0answers
56 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 ...
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0answers
93 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 ...
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56 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 ...
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0answers
21 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 ...
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0answers
43 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 ...
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0answers
44 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 ...
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0answers
91 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 ...
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0answers
165 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
149 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?
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11 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
24 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
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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27 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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27 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 ...
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71 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 $OA_{i}A_{i+1}$ ...
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0answers
78 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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40 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 ...
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0answers
62 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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33 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
65 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
116 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 ...
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15 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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14 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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0answers
27 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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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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0answers
52 views

Difference between polyhedral, CSG and B-rep

I am working on the 3D object modeling project. I found objects can be represented in the form of Polyhedrol model, CSG (Constructive Solid Geometry) model, and as well as B-Rep (Boundary ...
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0answers
104 views

Finding the area of non-standard polygons missing measurements

How do you find the area of non-standard polygons missing a few measurements? Here's a replica of a polygon i had to find the area of on my 9th grade final exam. I understand the fact you have to ...
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0answers
38 views

“Fundamental region” for non-discrete Moebius groups.

Suppose we are given a discrete, faithful representation $\rho$ of $F_2=\langle a,b|\rangle$, the free group on two generators, into $\mathbb{P}SL(2,\mathbb{R})$, so that the quotient is homeomorphic ...
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49 views

Internal angle formula of a generalized polygon as a function of side length and apothem

I am looking to compute the internal angle of a generalized regular polygon (spherical, euclidean, or hyperbolic) as a function of its apothem and side length. I know the equation for a euclidean ...
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0answers
43 views

Geometrically, what is the difference between a “flat face” and a “non-flat” face?

I was curious when I was checking sites like MathisFun, and I came across a pretty unclear system that defines a "flat face" and as a "non-curving" face of a shape; a polyhedron. However, I have to ...
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0answers
56 views

Inward-pointing normal and co-ordinate systems

I'm doing a course in computer graphics, and as such, we're being taught measures on how to deal with the Hidden Surface Removal problem. One of the topics covered was "back-face detection", that is, ...
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21 views

Calculate self-avoiding-filling-polygons

Definition of self-avoiding-filling-polygon In Euclidean graph where each vertex is a point on the $2D$ plane, so the weight of each edge is the Euclidean distance between the vertices. ...
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27 views

Where are $A,B,C$ in the regular $n$-gon such that $\min (|AB|+|BC|,|BC|+|CA|,|CA|+|AB|)$ gives the max?

Let $F_n$ be the regular $n$-gon of edge-length $1$. Let us consider taking three points $A, B, C$ in $F_n$. Suppose that you can take a point on the edge of $F_n$. Supposing that $|AB|$ represents ...
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25 views

About the relation between two regular icosahedrons and a regular dodecahedron

Let $C$ be the regular icosahedron, each of whose vertex exists at the centroid of the each surface of the regular dodecahedron $B$, each of whose vertex exists at the centroid of the each surface of ...
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73 views

First event in a straight skeleton

Is there a simple geometric criterion to check whether the first event in (the wave propagation of) a straight skeleton is an edge event or a split event? The literature I could find is computational ...
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0answers
59 views

Finding every $(m,n)$ such that a regular $n$-gon is inscribed inside a regular $m$-gon.

I found the following five types. 1. $(m,n)=(kn,n)$ for $n\ge3, k\ge1$. 2. $(m,n)=(hk,2k)$ where $h\ge1$ is an odd number and $k\ge2$. 3. $(m,n)=(m,3)$ where $m\ge4$ and $m\not\equiv0$ (mod $3$). ...
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90 views

Solving the “Library of Babel” puzzle, but for polygons.

The Library of Babel is a story about a universe whose contents are every possible 410-page book that could possibly exist. After a conversation with someone about doing this with images, and coming ...
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0answers
89 views

Straight skeleton is a tree

Can anybody give me a hint on how to prove that the straight skeleton of every polygon is a tree. Here is the definition of the straight skeleton (taken from Wikipedia): The straight skeleton of a ...
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0answers
175 views

k-dissection of a polygon with non-intersecting diagonals

I am trying to use the vertex coalescing method like the one mentioned here, page 10, to count: Number of dissections of a polygon using non-intersecting diagonals into even number of regions. I am ...
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0answers
179 views

Area of a simple random quadrilateral

Given four randomly chosen points with known coordinates, how to compute the area of a not self crossing quadrilateral? There is a formula if the points are ordered (direct or indirect). So the ...