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Questions tagged [plane-geometry]

Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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Pseudo-Droz-Farny circles

I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
A.Zakharov's user avatar
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A center of convex planar regions based on chords

This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions. A point $P$ in the interior of a planar convex region $C$ divides ...
Nandakumar R's user avatar
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A generic question on circles associated with a triangle

This question is inspired by two posed by P.Terzić (both given elegant synthetic proofs by F. Petrov). The starting point is a triangle $ABC$ and a triangle centre $G_1$. There are two classical ...
bathalf15320's user avatar
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Length of isoline $x(1-x)y(1-y)=c$

For the integral appearing in this answer, it may be beneficial to derive the length $L(c)$ of the isoline: $$x(1-x)y(1-y) = c,$$ where $x,y$ are ranging in $[0,1]$, and constant $c\in [0,\frac1{16}]$....
Max Alekseyev's user avatar
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How can construct the equilateral $A''B''C''$ such that area of $A''B''C''$ is biggest

Let $ABC$ be arbitrary triangle in a plane. Let $A'B'C'$ and $A''B''C''$ be two equilateral triangles such that $A \in B'C'$, $B \in C'A'$, $C \in A'B'$ and $A \in B''C''$, $B \in C''A''$, $C \in A''B'...
Đào Thanh Oai's user avatar
1 vote
1 answer
227 views

On comparing planar convex regions of equal perimeter and area

Definitions: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set. Given two planar convex regions $C_1$ ...
Nandakumar R's user avatar
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Complexity of tour-expansion heuristic for the planar Euclidean TSP

This is a followup question to this one: Computational Geometric Aspects of Greedy Tour Expansion. Assume that the candidate point, whose insertion into current incurs the least tour-length increase, ...
Manfred Weis's user avatar
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Geometric interpretation of metric [closed]

For metric in $\mathbb R^2$ (polar coordinates) Pythagorean triangle elements can be visualized in a differential form as seen at top yellow triangle. This happens for metric: $$ ds^2= dr^2 +(r d \...
Narasimham's user avatar
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Trapping lightrays under nonstandard reflections and/or paths

Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open: "It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...
Joseph O'Rourke's user avatar
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On non-convex polygons that tile convex polygons

Polyominoes are rectilinear polygons - all angles are 90 or 270 degrees. It is well-known that among non-convex polyominoes are several whose copies can neatly tile rectangle. And for several such '...
Nandakumar R's user avatar
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Some Problems On Apollonian Gasket

Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems: Let three $(A)$, $(B)$, $(...
Đào Thanh Oai's user avatar
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Comparing Different Notions of Unicoherence in the Plane

Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...
John Samples's user avatar
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Patterns in local winding number sequences

This is something of a followup to an earlier question Can any sequence of consecutive integers be realized as winding numbers?, whose answer is Yes. Now I would like to define a local winding number ...
Joseph O'Rourke's user avatar
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When do quasiperiodic functions on $\mathbb{R}^2$ have an average over the plane?

For example, consider the following \begin{equation} \lim_{T\rightarrow\infty}\frac{1}{(2T)^2}\int^T_{-T}\int^T_{-T}\big[\cos(v/a)-\cos(u/b)\big]\cos(\sqrt{u^2+v^2})\ du\ dv, \end{equation} where $...
Omarco's user avatar
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Determine sub-polygon from line segments with known member connectivity

Test Polygon: Consider the following polygon as attached. Let the known parameter be as follows: •Member to member connectivity, i.e. it is known that A – B, X – E, F – B, etc. for all the members. •...
sidb's user avatar
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Generalization of Ellipse via Fixed Sum of 3 Distances to "Foci"

It is a well known fact, that ellipses can be defined as $$\{x\in\mathbb{R}^2\ \ |\ \ \|x-A\|+\|x-B\|-\|B-A\|=e\in\mathbb{R}_0^+;\ A,B\in\mathbb{R}^2\}$$ Question: has the generalization $$\{x\...
Manfred Weis's user avatar
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Intersection points of closed curves inscribed in a convex polygon

Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...
Alan Horwitz's user avatar
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Non-Convex Polygons with "Antipodal Visibility"

by "antipodal visibility" of planar, simple polygons I mean the following property: if two points $p$ and $q$ on the polygon's boundary divide the polygon's boundary into two polylines of equal length,...
Manfred Weis's user avatar
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Which plane convex arcs have the smallest maximum curvature?

Let $p$ and $q$ be positive real numbers with $p \leq q$. Suppose that $H(p,q)$ is the class of all convex arcs $c$ in the Cartesian $x-y$ plane which satisfy the following conditions: (1)The $y$-...
Garabed Gulbenkian's user avatar
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2 answers
129 views

Planar curves identical to their inverses

Is the right strophoid the only planar curve $C$ whose inverse curve w.r.t. some circle (in this case: centered on the origin) is identical to $C$?               &...
Joseph O'Rourke's user avatar
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1 answer
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Hyperbolic version of Sylvester co-linear problem

Is the hyperbolic version of Sylvester co linear problem true?
Ali Taghavi's user avatar
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1 answer
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Does anyone know of an academic reference for this proof that the tangent to a hyperbola at a point bisects the angle between each focus to the point?

I am writing a report and as part of it I need to prove the property that for a point $P$ on a hyperbola, the tangent to the hyperbola at $P$ bisects the angle $\angle F_1PF_2$ where $F_1$ and $F_2$ ...
Louisa's user avatar
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3 answers
115 views

Calculating radii allowing for circular placement of polygonal linkage's joints

Given a planar polygonal linkage defined by a sequence of $n$ hinge joints $(j_0,\,\cdots,\,j_{n-1},j_n = j_0)$ with links of fixed lengths $\lbrace\|j_{k+1}-j_k\|=d_k\ |\ 0\le k\lt n\rbrace$ between ...
Manfred Weis's user avatar
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2 answers
3k views

Determine the boundary points of a set of points [closed]

I have a set of points $S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$. Then how to find the boundary points (which is a subset of $S$) of $S$? There are methods like convex hull, concave hull and $\...
janak's user avatar
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1 answer
332 views

Finding the minimum-area parallelogram containing all vertices of a cuboid projected to a plane

Description edited after comments: Let's say I have a 3d cuboid that I projected onto a 2d plane. Now I want to find the minimum-area parallelogram that contains all those projected vertices. How do ...
Moody's user avatar
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2 answers
130 views

Algebraic planar curve with precisely $n$ closed components? [closed]

For each integer $n$ I am looking for a real-valued polynomial in two variables, $A_n(x,y)$, such that $A_n(x,y) = 0$ defines a curve with precisely $n$ closed components in the plane $\mathbb{R}^2$. ...
jess's user avatar
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2 answers
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Why does the area function of a parallelogram have a nonintuitive geometric solution?

I was reading a blog post on a simple derivation of the cross product. I learned how to determine the area of a parallelogram enclosed by two vectors $A$ and $B$. First, here is the proof of the ...
Shaun Lebron's user avatar
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2 answers
177 views

Radical line of two ellipses

The equation of an ellipse with focal points $(a_1,b_1)$ and $(a_2,b_2)$ which passes through the point $(a_3,b_3)$ is given by the equation $$\begin{gathered} \sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-...
Benjamin L. Warren's user avatar
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1 answer
126 views

On 'special' points on uniform planar convex regions defined in terms of moment of inertia

The following can be easily proved using perpendicular axes theorem and intermediate value theorem: Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least ...
Nandakumar R's user avatar
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Extending functional inequality from rectangles to parallelograms

Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ satisfying $f \geq 0$, $f(0,0) = 0$, $\frac{\partial{f}}{\partial{x}} \geq 0$, $\frac{\partial{f}}{\partial{y}} \...
Charles Pehlivanian's user avatar
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1 answer
1k views

Fast way to generate random points in 2D according to a density function

I'm looking for a fast way to generate random points in 2D according to a given 2D density function. For instance something like this: Right now I'm using a modified version of "Poisson disc&...
shoosh's user avatar
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0 votes
1 answer
176 views

Expected area of a pentagon formed from a randomly broken stick [closed]

Suppose we break a stick of length one at four randomly and independently chosen points and that the resulting pieces form a pentagon. Such a pentagon can be formed with probability $1-(5/16) = {11\...
John Smith's user avatar
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0 answers
44 views

Lattice points in the boundary of a Minkowski sum of two convex lattice polygons

Let $P$ and $Q$ be two convex lattice polygons in $\mathbb{R}_+^2$ and let $P+Q$ be their Minkowski sum. Given a set $S \subset \mathbb{R}^2$, we let $L(S) =\#( S \bigcap \mathbb{Z}^2)$. The equality $...
Yromed's user avatar
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0 answers
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Coordinates of the centers of the insphere and circumsphere

Suppose we are working in $\mathbb{R}^3$ space and we have four non-coplanar and non-collinear points, $(x_a, y_a, z_a)$, $(x_b, y_b, z_b)$, $(x_c,y_c, z_c)$, and $(x_d, y_d, z_d)$. How does one ...
Benjamin L. Warren's user avatar
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1 answer
231 views

Divide angles by coefficients relate to Fibonacci sequence

In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle $(...
Đào Thanh Oai's user avatar
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0 answers
41 views

Air-transmissive mirror (illumination problem for a wall)

I propose an alternative version to the illumination problem (where the mirrored walls surrounding a room prevent light from reaching some region). Here, our building area is an infinite straight band ...
Bence Hervay's user avatar
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In how many ways is it possible to order the sides and diagonals according to their length for all n-gons?

If we'd do it for example for 4-gons, for quadrilaterals, we could start with all the possible quadrilaterals. We could say that the four vertices are a,b,c and d. And then we'd have 6 lines, I mean, ...
Dr.X's user avatar
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0 answers
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A surprising result with the Riccati difference equation

I was looking at the Riccati difference equation with positive and negative indices $$ R_n=\frac{aR_{n-1}+b}{cR_{n-1}+d}\quad n\in[0,N]\\ R_n=\frac{-dR_{n+1}+b}{cR_{n+1}-a}\quad n\in[-N,0]\\ $$ along ...
Cye Waldman's user avatar
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0 answers
128 views

Concurrencies determined by intersections of angle trisectors (and isogonal lines) in a triangle

The famous Morley’s theorem, states that in a triangle the interior angle trisectors, proximal to sides respectively, meet at the vertices of an equilateral. However the six trisectors meet at 12 ...
Spiridon Kuruklis's user avatar
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1 answer
55 views

On 'axiality' of planar convex regions

Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry) Consider an alternative definition of axiality as follows: For a convex region C, consider a ...
Nandakumar R's user avatar
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Vertex configuration to tile repeat unit

I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
Jim Z's user avatar
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0 answers
610 views

How to find a point on a line that minimizes sum of distances from three given points?

Let there be given three points $x_1, x_2, x_3$ and a line $l$ on a plane. Does there exist an explicit method of finding such a point $p$ on $l$, that sum of distances of $p$ from $x_1, x_2, x_3$ is ...
NikoWielopolski's user avatar
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0 answers
36 views

What is the locus defined by those equations?

I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by $\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$. I know that if $\...
MysteryGuy's user avatar
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0 answers
167 views

Infinity new equilateral triangles in one configuration of triangle plane

An equilateral triangle constructed from a reference triangle is a topic which is intersested by plane geometry lovers. See Napoleon equilateral triangle, Morley equilateral triangle....In this topic ...
Đào Thanh Oai's user avatar
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0 answers
89 views

What is the minimal number of lines needed to partition a simplex into cells of diameter at most $\epsilon$?

I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines ...
User123321's user avatar
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0 answers
332 views

Are finite projective planes isomorphic to PG(2,q), where q=p^k with p a prime?

I can't find any reference to the following question: Are finite projective planes isomorphic to $PG(2,q)$, where $q=p^k$ with $p$ a prime? I think it's meaningless to continue studying without ...
Frederick Silva's user avatar
-1 votes
2 answers
240 views

Locus of points for which the sum of the angles subtended there by two different line segments is a constant

Given a line segment AB, the locus of points P such that the angle APB has a constant value is a 'biconvex lens' formed by 2 circular arcs that passes through the points A and B. Special case: if APB ...
Nandakumar R's user avatar
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-1 votes
1 answer
824 views

Johnson-Lindenstrauss lemma preserves angles

Edit (April 1, 2020): I formalized the original question (from the reference below) wrong. Please see @AndreasBlass correction. Also, all the suggestions proposed so far doesn't seems to be related ...
keyboardAnt's user avatar
-2 votes
1 answer
587 views

Is the conjecture true for n-sphere $(n>2)$? [closed]

This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here: PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...
Đào Thanh Oai's user avatar
-3 votes
2 answers
546 views

Hexagon Formed by connecting Trisections of triangle sides [closed]

Is there a theorem for the area of the hexagon formed by connecting the points formed when the sides of a triangle are trisected? It appears that the ratio of the area of the triangle to the area of ...
Michael Pruner's user avatar

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