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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 polygon that traverses all given points with minimal circumference but has a single surface

I know about Convex Hull, but convex hull is convex, which isn't what I want. I want a polygon that will always try to minimize it's perimeter however will not have multiple surfaces: For example, I ...
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15 views

Angles between vectors of complex polygons

currently I try to get an insight into the following problem. Consider a (closed) complex polygon (edges can intersect) consisting of $n$ vectors. For simplicity, assume that every vector has unit ...
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The relationship between the median,sides and the height of a triangle

Let $BC = a$, $b = CA$, $c = AB$ be side lengths in triangle $ABC$. Indicate the $h$ length of the height against the side $AB$ and the $m$ the length of the median from the corner $C$. Below are four ...
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Every $(2n)$-gon have a diagonal which isn't parallel to any side of this polygon

Prove, that every $(2n)$-gon have a diagonal which isn't parallel to any side of this polygon. I was thinking about sth like: Let's suppose that ($2n-3$) diagonals from one point are parallel with ($...
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1answer
107 views

Area of Generalized Koch Snowflake

In the Koch snowflake, the zeroth iteration is an equilateral triangle, and the n-th iteration is made by adding an equilateral triangle directly in the middle of each side of the previous iteration. ...
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1answer
53 views

parametrical representations of polygons

Could you please explain, how one gets this Parametric representation of a solid trapezoid ? I mean the procedure and not the answer. I have some linear geometry (as polygons), and I need to represent ...
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1answer
27 views

Angle created by three distincts random vertices

Assume you have a regular polygon ( $n$-sides). and Let $A=\{ x_0, x_2, \cdots , x_{n-1} \}$ be vertices of the polygon. My Question is: Are there is any formula that tell us what is the angle ...
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21 views

Equivalence between polygon definitions.

I generaly find the definition of polygon as a "polygonal chain", but I barely find another definition that defines a convex polygon as the intersection of a finite number of half-planes and defines ...
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1answer
13 views

Is the average number of edges per 2-faces of a convex 3-polytope always below six?

Is the average number of edges per 2-faces of a convex 3-polytope always below six? Which theorem can answer this question or how do you prove this?
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1answer
40 views

Caculate area of polygon in 3D [closed]

I have a list vertex with 3D coordinate of a polygon. How to canculate its area. Should I convert it into 2D?
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0answers
33 views

Reconstructing a polygon from the Midpoints of Its Sides

I was reading through Dijkstra's 'A Collection of Beautiful Proofs' and stumbled upon this elegant piece of work: 11. Reconstructing an odd polygon from the midpoints of its sides. We shall ...
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1answer
55 views

How to find the number and coordinates of self-intersections points for a polygon?

I have a self-intersecting polygon defined by $n$ points on the plane: ...
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1answer
38 views

Regular polygons with equal height

Sorry for my naivety I am trying to plot the points of n sided regular polygons but maintaining the height between odd and even sided polygons. Is there a sensible algorithm for doing so before I ...
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1answer
38 views

Finding integer coordinates for a pentagon, hexagon, heptagon, octagon, and nonagon, etc.

Wondering what the formula is for finding integer coordinates for an arbitrary "regular" polygon. By regular I mean symmetrical polygons like pentagon, hexagon, etc. In particular, I would like to ...
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1answer
39 views

How to rigorously show that elementary operations on polygonal presentations yields homeomorphic spaces?

In Lee's book Introduction to Topological Manifolds, he discusses elementary operations on polygonal presentations. Before the question, here are the terminologies that I am going to be using: A ...
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Products of k/l-gons

For $k \geq 3$ let $P_k = conv\{(\cos\frac{2\pi\cdot i}{k}, \sin\frac{2\pi\cdot i}{k})\ |\ 0 \leq k < i\}$ be a regular $k$-gon in $\mathbb{R}^2$. We want to look at product $P_k \times P_l$ in $\...
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2answers
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Complex coordinates of the vertices of a polygon

If $z_0$ be the centre of a regular polygon of $n$ sides and $z$ be its one vertex $A_1$, then the vertices $A_2, A_3,\dots, A_n$ (proceeding in anticlockwise direction, taking actual position of ...
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Finding the centre of the largest inscribed circle in an irregular polygon

[edited title] I need to find A centre point of an irregular polygon. Being an irregular polygon, I'm aware that there can be no geometric centre. I have a very specific shape in mind, it is a ...
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1answer
43 views

Determine if a point lies in a quadrangle [duplicate]

I have a quadrangle which sides consist of parts of rays, and I only know the coordinates of two points on each ray. I need to determine if a point $(x,y)$ lies in such quadrangle. In this picture, ...
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1answer
24 views

Number of ways to choose a closed path of given length on a square lattice

Also known as self-avoiding polygons, this is an unsolved problem. However, to leading order in the asymptotic limit, the number of polygons of a given perimeter scales exponentially with perimeter ...
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1answer
74 views

Fourier series of regular polygons

The definition of a regular polygon by two real-valued functions $(x(t)$, $y(t))$ – or alternatively by a complex-valued function $x(t) + iy(t)$ – suggests to calculate the Fourier series $...
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1answer
39 views

What does the shoelace formula mean for polygons with crossings?

Given a simple polygon with vertices (in order) $v_1,v_2,\ldots,v_n$, the area of this polygon can be computed based on only the coordinates of these vertices via the shoelace formula: $$A=\frac{1}{2}...
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1answer
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Systematic approach to triangulation closed combinatorial surfaces

I was wondering whether there is a systematic approach to the triangulation of closed combinatorial surfaces, which we know can be shown to be homeomorphic to polygons with complete set of side ...
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2answers
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What is the length of $x$ in this pentagon diagram?

ABCDE is a regular pentagon. $\angle AFD = \angle EKC$ $|FH|=1$ cm; $|AH|=3$ cm What is $|DK|?$ I know that triangles $EFA$ and $DEK$ are similar and that $|EK|=4$ cm. Also because this is ...
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Rotate and Translate Irregular Polygon to X-axis

I need to rotate and translate an irregular polygon so that a chosen edge is on the x-axis and the "inside" of the polygon is above the x-axis. I know how to translate and rotate a polygon using a ...
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0answers
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Finding the vertices of a regular n-sided polygon using its centroid

How do you find the vertices of a regular $n$-sided polygon using its centroid, with the knowledge that each vertice is distance $d$ from the centroid.
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1answer
31 views

Calculate Area of 4 points polygon from distance between poins

I have 4 points on ground, something like what is depicted in this sample: https://www.mathopenref.com/coordpolygonarea.html The problem is that I don't know how to determine the coordinates of my ...
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2answers
57 views

Area of a quatrefoil inside a circle

What is the maximum area of a quatrefoil that is inscribed in a circle of radius 6? My first guess was to cut the circle into small regions but that doesn't seem to work. The solution does not have ...
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1answer
27 views

Polygons with two different angles [closed]

I have the following questions: Let us consider polygons with only two different interior angles $\alpha$ and $\beta = \pi - \alpha$. For which frequencies of these angles is it possible to form a ...
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1answer
47 views

Convex polygon inscribed in a circle with vertices that create matrix. Prove that the rank of this matrix is less or equal 2.

Consider a convex polygon inscribed in a circle with vertices $P_1$, ..., $P_n, \ n \ge 3$. Let $A$ be the matrix $n \times n$ such that $\begin{equation} a_{ij} = \begin{cases} |P_iP_j| ...
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2answers
81 views

Circle measurement of Archimedes

Let $f_n$ or $F_n$ be areas of the regular $n-$ polygon described to the unit circle or circumscribed. Show $f_{2n}=\sqrt{f_nF_n}$ and $F_{2n}=\frac{2f_{2n}F_{n}}{f_{2n}+F_n}$ In the solutions ...
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2answers
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What is the length of the side of a regular hexagon and why?

I think it is 1. Because from the Picture it Looks like that but can somebody explain me why this is so?
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1answer
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What is the symmetry point group of a regular n-gon, where n is even with the opposite vertices identified? Is it $D_n$? [closed]

If we have a regular 2n-gon, what is its symmetry group if we identify the opposite vertices (the ones that are on the same line). Do we get the group that is isomorphous to the group of the regular n-...
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2answers
59 views

How to find lengths and corner coordinates of an irregular pentagon

How do I find the side lengths and therefore corner coordinates of a pentagon with the following internal angles: A = 140°, B = 60°, C = 160°, D = 80°, E = 100° ?...
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How to compare centrality of points in different shaped and sized polygons

Suppose I have several polygons of varying shape and size, each containing a single point. I want to compare the centrality of each point, and determine which point is more central to its polygon. So ...
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1answer
47 views

How can I find the measure of every angle in a star polygon?

I'm unfamiliar with these kinds of problems. I looked up some formulas and it says for an $(n,3)$ family of star polygons, $\theta = \frac{(1-\frac{6}{n}}{180}$ How do I get these formulas and what ...
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1answer
28 views

Side length of a decagon given the circumradius, without trigonometry

How can I find the side length of a regular decagon, given a circumradius R? I know how to do it with trigonometry, is there another way using Ptolemy's Theorem? Thanks!
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1answer
66 views

Determine Direction of Normal vector of Convex Polyhedron in 3D

How can I determine direction(point inside or outside) of normal vector drawn on one side of the polyhedron? Known informations; coordinates of all corners in 3d as x,y,z which face of the normal ...
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How to create a convex polygon which is the superset of a given polygon?

I have a JTS_footprint like this, POLYGON ((46.542770 -65.500984,54.130459 -63.660416,59.213402 -66.664993,51.004562 -68.736069,46.542770 -65.500984)) How do I ...
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1answer
55 views

Squares on the sides of a regular pentagon

On each side of a regular pentagon a square lying outside the pentagon is constructed. (see the picture below) $X_1,..., X_5$ are the centers of the squares. $P$ and $Q$ are points of intersection ...
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3answers
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How would I find the lengths of the sides of a regular pentagram if given the perimeter of the circle that the pentagram is inscribed to?

For a Turtle program in Python that I am working on, I will need to draw out a star. This star is a regular pentagram, meaning that each of the sides are of equal length. As well, the pentagram is ...
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Intersection of isometries of a polygon

Suppose we have a two dimensional polygon $P$. And a sequence of polygons $(P_i)$, where each $P_{i+1}$ is a small translation/rotation of $P_i$. I am interested in situations where $\cap P_i$ is ...
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1answer
30 views

How to measure an angle in a polygon that is more than 180?

Assume we have an arbitrary polygon that has no holes nor self edge intersections, but can otherwise be concave and deformed. Assume the vertices are ordered either clockwise or ccw. So for example ...
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0answers
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How to pick a “center” of a concave polygon?

I asked a question on how to scale concave polygons and a couple of people suggested some very clever solutions. The issue is that these solutions rely on picking an appropriate point $C$ in the ...
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2answers
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How to scale a polygon such that all the points lie within the original polygon?

With a convex shape, like a circle, we can create a set of similar shapes, all contained within one another, by centering the shape at the origin and scaling it. So we can get the following: With a ...
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1answer
99 views

Centroid within non-convex 2d polygon

The centroid of an object is defined as the arithmetic mean of all points of the object. For non-convex objects, the centroid is often not a part of the object itself: Is there a definition of a ...
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1answer
30 views

Algorithm for converting a coordinate into angles of a pentagon.

I will go ahead and admit, this might just be something obvious but I did research and couldn't find anything. I have a pentagon, and I know two top points (A & B) and the distance between them (...
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0answers
25 views

The best approximation method to recover original polygon outline before rasterization procedure

I have a polygon, originally created as a Bézier Curve (black outline on the picture), and then saved as a polygon with enough points to call it smooth (at this scale). Then this polygon was ...
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2answers
2k views

If a regular polygon has a fixed edge length, can I know how many edges it has by knowing the length from corner to its center?

So I wonder if there is a formula so that when there's a defined edge length, I can calculate a regular polygon's edges amount by knowing its length from corner to center, or vice versa. So let's ...
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1answer
44 views

What is a non-concave and non-convex polygon called?

I am writing a software function to plot the outer points of an n-sided polygon and I'm having trouble ensuring I use the correct terminology. The function I've written simply renders the calculated ...