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
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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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17 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 ...
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1answer
32 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
33 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
61 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
27 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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17 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
21 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
15 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
25 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
39 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
65 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
37 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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29 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
55 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
45 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 ...
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1answer
22 views

I need help doing a basic geometry problem

I think it can be solved using parallels but I have no idea how. Solve for "X"
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2answers
47 views

What's the fewest number of sides required to make a polytope in n dimensions?

In 2 dimensions it takes at least 3 sides to make a polygon, the triangle, and in 3 dimensions it takes at least 4 faces (so far as I'm aware) to make a polyhedron. Can this rule be generalized to ...
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18 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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1answer
133 views

Constructing a regular right angled hyperbolic hexagon

I would like to construct a regular right angled hexagon in a klein model. I'm having a hard time understanding why this method works, here is what my professor did in class. Any additional comments ...
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1answer
34 views

Determine missing angle in polygon

I'm trying to figure out this question: Determine the measure of angle a I'm guessing $a=96\unicode{0186}$ using the following work: $$a = 180 - 84 = 96 $$ ...
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2answers
39 views

Bisectors of adjacent angles of a parallelogram meet on midline?

Suppose $KLMN$ is a parallelogram, and that the bisectors of angle $K$ and angle $L$ meet at point $A$. Prove that $A$ is equidistant from $\overline{LM}$ and $\overline{KN}$, without using ...
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26 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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1answer
18 views

Minkowski difference of two convex polygons

I just want to make sure that the following algorithm is correct for computing the Minkowski difference of two shapes $A,B$: $\text{Minkowski}(A,B) = \text{ CH } \{x: x = a - b \text{ for } a \in ...
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73 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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1answer
27 views

Is there an efficient, general formula to verify if a number is a n-sided polygonal number?

I've seen formulas to verify if a number is a triangular number, a pentagonal number, or a hexagonal number, but I haven't seen a general formula for verifying if a number is an n-sided polygonal ...
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1answer
65 views

Calculating perimeter of n-sided regular polygons using only height?

Say you were drawing n-sided regular polygons on a square grid (first side you drew being flat always). You wanted any polygon you drew to be 100 units high, aka the uppermost point being y = 100. ...
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33 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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4answers
70 views

Proof that the number of diagonals of a polygon is $\frac{n(n-3)}{2} $

For $n \geq 3$ proof that the number of diagonals of a polygon is $\frac{n(n-3)}{2} $ using induction. I don't know how to start this problem, can you give me a hint?
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20 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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1answer
27 views

Relation for hyperbolic pentagon.

I am trying to get a relation between the length of the sides and the angles of a hyperbolic pentagon. In literature I can find relations for pentagons which has at least three Right angle. So my ...
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40 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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2answers
63 views

Polygon and Pigeon Hole Principle Question

Seven vertices are chosen in each of two congruent regular 16-gons. Prove that these polygons can be placed one atop another in such a way that at least four chosen vertices of one polygon coincide ...
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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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2answers
61 views

Show that diagonals intersect at common point

Given is octagon where opposite sides are equal length and parallel. Show that diagonals: $AE,DH, BF, CG$ intersects at point $S$ So I have tried to create a parallelograms $AHED$ and $BCFG$ and use ...
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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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41 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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1answer
52 views

Symmetry center of hexagon

How to show that the figure has a symmetry center for instance if we have a convex hexagon where opposite sides are of equal lenght and parallel?
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43 views

What kind of shape?

To construct this shape, draw a circle. Place the compass on a point on the circle and draw an arc of the same radius as the circle. Now place the compass at the intersection of the arc and the circle ...
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2answers
264 views

Algorithm of cutting a polygon into equal parts

I have a convex polygon. I need to divide it into 4 equal parts using the two slit. For example, if I have a square, I have to cut it along the diagonals. Are there some common algorithm for this ...
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4answers
50 views

Geometry terminology: concrete vs. continuous polygons?

I am trying to find the proper terminologies for 2 kinds of shapes: The first type of shape I'm calling "concrete polygons". They have a finite number of straight sides (connecting at vertices) and ...
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1answer
50 views

Bouquet of Two Circles and Based covers

I need help with the following: Let $B$ denote the bouquet of two circles labeled by $a,b$. We regard $\pi_{1}(B)\cong \langle a,b | -\rangle $ where the basepoint of $B$ is its vertex, and the two ...
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32 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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1answer
55 views

arc length of inscribed polygons with n=1,2 and 4 equal sides curve y=x(4-x)^(1/3)

Consider the curve y = x (4-x)^(1/3), 0< x <4 (a) Compute the lengths of inscribed polygons with n = 1, 2, and 4 sides. (In other words, divide [0,4] up into 1,2, and 4 equal segments, and ...
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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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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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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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1answer
139 views

Expressing vectors in an octagon

I'm having trouble with this question in my course. I am to consider a regular octagon with vertices A, B, C, D, E, F, G and H in counter clockwise order. The vectors $\overrightarrow{AC}$ and ...
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1answer
19 views

COM of a polygon

The COM of a point is the point itself. With another point you can construct a line. The COM of a line is 1/2 the distance between the COM the old point and the new measured form the former, with ...
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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 ...