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

Filter by
Sorted by
Tagged with
3 votes
1 answer
343 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 $B_n$? $$B_n=\sum_{1\le{i}\...
mathlove's user avatar
  • 4,727
11 votes
2 answers
448 views

Metrically homogeneous subsets of the plane

A metric space $M$ is metrically homogeneous if for every pair of points $x, y \in M$ there is an isometry $f$ of $M$ onto $M$ such that $f(x)=y$. What is known about metrically homogeneous spaces? ...
Wlodek Kuperberg's user avatar
8 votes
1 answer
608 views

Small quadrilaterals containing a given convex region

It is easy to prove that (*) Every convex planar set of area 1 is contained in a quadrilateral of area 2. It is also easy to see that statement (*) remains true if the constant 2 is replaced with a ...
Wlodek Kuperberg's user avatar
6 votes
2 answers
381 views

Solving for special rational triangles

I ran into a need for isosceles triangles that (1) have the two equal integer side lengths $a$ (but the base $x \in \mathbb{R}$), and (2) the apex angle $\gamma$ is a rational multiple of $\pi$. &...
Joseph O'Rourke's user avatar
31 votes
2 answers
2k views

Tiling of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes, that is, it is a curve of constant width. (The width between two parallel tangents to the curve are independent of the ...
Per Alexandersson's user avatar
6 votes
1 answer
665 views

Does an origin-centered ellipse in the plane intersect each $L^p$-circle at most 8 times?

The question is in the title: Let $E$ be an origin-centered ellipse in ${\mathbb R}^2$ and let $S$ be an "$L^p$-circle": $S = \{(x,y) : |x|^p + |y|^p = \text{const}\}$, where $1 \leq p \leq \infty$. ...
Ryan O'Donnell's user avatar
5 votes
4 answers
23k views

How to find overlap between two convex hulls, along with the overlap area

I have two boundaries of two planar polygons, say, B1 and B2 of polygons P1 and P2 (with m and n points in Boundaries B1 and B2). I want to find out if the polygons overlap or not. If they overlap, ...
Harsha's user avatar
  • 53
5 votes
2 answers
365 views

Diameter-area ratio for affine tranformations.

Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$? I found only one reference, to "Über einige ...
Nikita Kalinin's user avatar
16 votes
3 answers
2k views

Applications of visual calculus

Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool. The basis is Mamikon's theorem. The area of a tangent sweep is equal to the area ...
2 votes
2 answers
465 views

Reference question: Poncelet theorem

A very famous theorem of Poncelet states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in ...
18 votes
5 answers
2k views

Definition of area

I am looking for an attractive, but rigorous definition of area; say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...
Anton Petrunin's user avatar
3 votes
1 answer
276 views

What is the sequence that maximizes this distance?

I have posted this question here without answer. Maybe I can get some light here. Suppose we are given $n$ segments $l_1,...,l_n$ in $\mathbb{R}^2$ such that $|l_i|=i,\ \forall\ i=1,...,n$, where $|...
Tomás's user avatar
  • 71
8 votes
2 answers
1k views

Spanning trees of plane graphs containing an edge of every face

I feel sure this must be known, but can I find it?? Which connected plane graphs (graphs drawn in the plane without crossings) have a spanning tree such that at least one edge of each face is in the ...
Brendan McKay's user avatar
17 votes
1 answer
453 views

The sparsest planar net that captures every unit segment

Let $\cal C = \lbrace C_i \rbrace$ be a collection of rectifiable curves in the plane with the property that every unit-length segment meets at least one curve in at least one point. Call such a ...
Joseph O'Rourke's user avatar
7 votes
2 answers
662 views

Does list of distances define points uniquely?

There are N points on a plane. Is it feasible to reproduce their relative location having only the list of distances. Assuming that translation, rotation and mirror are allowed in the result. The ...
janst's user avatar
  • 73
0 votes
0 answers
330 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
3 votes
1 answer
362 views

On distances between points on the plane

Take a set of $2n$ points in the plane and assume that no open set of diameter $1$ contains more than $n$ of these points. Question: can we pair up the points so that the distance between the points ...
TOM's user avatar
  • 2,218
5 votes
2 answers
663 views

Recognize this plane curve?

An aspect of my work led to a plane curve with implicit equation $$ x^2+y^2 = 3 (y/2)^{2/3} + 1 $$ Actually, I started with the parametrization below and derived from it the equation above: \begin{...
Joseph O'Rourke's user avatar
8 votes
2 answers
740 views

Are point sets of the same order type connected by continuous (order type)-preserving motion?

Given two general position point sets in $\mathbb{R}^2$ of the same size and order type, is it possible to continuously move the points of one set until they coincide with those of the other set in ...
Nima Hoda's user avatar
2 votes
0 answers
155 views

A relation on triplets of points in the plane

This question is a follow up of my previous one (Planar sets closed under intersection of circles, Planar sets closed under intersection of circles) and is motivated by G. Zaimi's answer https://...
Denis Serre's user avatar
  • 51.5k
4 votes
1 answer
317 views

Planar sets closed under intersection of circles

Let $P$ be the plane with a point at infinity. By plane, I mean the Euclidian plane, and therefore it has circles. A line is also a circle, though its center is at infinity. If $A\subset P$ has ...
Denis Serre's user avatar
  • 51.5k
3 votes
2 answers
1k views

maximum number of shortest path among a set of n triangle obstacles

Assume that we have a two distinct points. The number of shortest path between these two points is one. When we add a triangle obstacle to the plane and this triangle intersects the line connecting ...
user23354's user avatar
32 votes
8 answers
4k views

Can Morley's theorem be generalized?

Morley's theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. In a talk some years ago, David Rusin made the provocative ...
Timothy Chow's user avatar
  • 78.1k
14 votes
3 answers
2k views

Optimal wireframe sphere

Suppose you have a length $L$ of metal pipe at your disposal, and you would like to build a wireframe unit-radius sphere, by bending, cutting, and welding the pipe into a connected structure $F$. Your ...
Joseph O'Rourke's user avatar
8 votes
2 answers
1k views

Quadrature of the Lune

What is a good reference for the following result which I believe is proved by Tchebotarev. There are exactly 5 types of Lunes that are squarable. (Hippocrates produced three and then two more were ...
Chebolu's user avatar
  • 575
6 votes
0 answers
148 views

Minmax problem for polygons

Let $\text{Pol}_n$ be the set of all convex polygons on a plane with $n$ vertices. For $P\in \text{Pol}_n$ denote by $\text{Tr}(P)$ the set of all triangles which vertices are some vertices of $P$. I ...
Norbert's user avatar
  • 1,687
10 votes
1 answer
375 views

Reorienting a ladder among $\mathbb{Z}^2$ poles

Imagine an object, which I'll call a ladder $\cal{L}$, a "racetrack" shape composed of a rectangle of length $L$ capped at either end by semicircles of radius $r$; so it is $L+2r$ tip-to-tip. View ...
Joseph O'Rourke's user avatar
9 votes
0 answers
1k views

Interpolating points with minimum curvature constraint

I have $n$ points $p_i$ strictly interior to a rectangle $R$, and I would like to connect them with a curve $C$ whose curvature is as low as possible. Let $\kappa_\max(C)$ be the sharpest (largest ...
Joseph O'Rourke's user avatar
25 votes
4 answers
1k views

Pinball on the infinite plane

Imagine pinball on the infinite plane, with every lattice point $\mathbb{Z}^2$ a point pin. The ball has radius $r < \frac{1}{2}$. It starts just touching the origin pin, and shoots off at angle $\...
Joseph O'Rourke's user avatar
11 votes
1 answer
706 views

Polygons uniquely inducing arrangements

A beautiful, relatively recent result is that, Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$. In a simple arrangement, every pair of lines intersect ...
Joseph O'Rourke's user avatar
2 votes
1 answer
397 views

An equivalence relation on the power set of the plane.

Let $R\subseteq\mathbb{R}^2$. Consider the set of all "horizontal sections" $H_R =$ {$ Rb|b\in\mathbb{R}$} where $Rb=${$ a\in\mathbb{R} | (a,b)\in R$}. Similarly consider the set of "vertical sections"...
Rogelio Fernández-Alonso's user avatar
15 votes
1 answer
810 views

Ratio of circumscribed/inscribed $(n{-}1)$-gons

As a discrete analog of the MO question, "Löwner-John Ellipsoid: incribed and circumscribed," I've been wondering what might be the maximum ratio of this quantity? Let $P$ be a convex ...
Joseph O'Rourke's user avatar
2 votes
1 answer
219 views

Maximal sets of algebraic curves, closed under rotation, dilation, and translation, that pairwise intersect at most twice

Consider a set of nontrivial algebraic curves on the plane groovy if that set is closed under rotation, dilation, and translation, and has the property that no two members of the set intersect more ...
Darren Ong's user avatar
3 votes
0 answers
190 views

Characterizing curves that bound strictly convex regions

Consider a closed curve on the plane so that if we perform any translation and dilation on it, the resulting curve intersects the original curve at most twice. Does this property characterize the ...
Darren Ong's user avatar
2 votes
1 answer
307 views

Connecting tangents of convex curves: at some point orthogonal?

Let $a(t)$ and $b(t)$ be two smooth, nested convex curves in the plane, $t\in[0,1]$:       Suppose the parametrization of $a()$ and $b()$ is such that $\dot{a}(t)$ is ...
Joseph O'Rourke's user avatar
5 votes
3 answers
2k views

Original proof of Pappus' Hexagon Theorem

Does anyone know where I can find an English translation, preferably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of Pappus' hexagon ...
Avi Steiner's user avatar
  • 3,031
10 votes
5 answers
1k views

Convex curves with many inscribed triangles maximizing perimeter

A classical nice result of Euclidean geometry states that the triangles maximizing the perimeter among all inscribed triangles of a given ellipse constitue a one-parameter family. Precisely, for each ...
Pietro Majer's user avatar
  • 56.5k
17 votes
2 answers
945 views

Isoperimetric-like inequality for non-connected sets

The classical isoperimetric inequality can be stated as follows: if $A$ and $B$ are sets in the plane with the same area, and if $B$ is a disk, then the perimeter of $A$ is larger than the perimeter ...
Guillaume Aubrun's user avatar
10 votes
1 answer
463 views

Chord arrangement that avoids confining small or large disks

These two questions are two-dimensional variations on this recent MO question, "Threading pinholes in the wall of cylinder to pass through an internal coordinate." Noam Elkies suggested that even a 2D ...
Joseph O'Rourke's user avatar
13 votes
1 answer
684 views

Including a Jordan arc into a Jordan loop (Can the Magi go home by another way?)

The title refers, of course, to Matthew (2:12) ''And being warned in a dream not to return to Herod, they departed to their own country by another way''. To be honest, it is not that specific ...
Pietro Majer's user avatar
  • 56.5k
4 votes
1 answer
256 views

Polar interpretation of convexity

Let $C$ be a convex polygon in the plane containing the origin, and let $r(\theta)$ for $\theta\in[0,2\pi)$ be a parametrization of its boundary. Is there a condition on $r$ that is equivalent to (or ...
Jennifer Gao's user avatar
6 votes
3 answers
2k views

Dissecting a square

Edited - some comments may now be out-of-date. I thought I had a complete set of solutions to this: ...
Colin D Wright's user avatar
2 votes
5 answers
6k views

Quadrilateral from 4 random points

Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points? I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q. But ...
roadrunner66's user avatar
5 votes
2 answers
649 views

Area of intersection of a family of circles in the plane

Suppose you are given a family F of circles in the plane such that each circle has radius 1. Let G be the family of circles with same centers as in family F but now each circle has radius $r$. Let A ...
csguy's user avatar
  • 51
4 votes
1 answer
457 views

Cubic curve closest to the given set of points

Assume we are given the set $S$ of $n$ points on the real plane and want to draw a parametrized cubic curve (actually a segment of Bézier spline) with fixed startpoint in such a way, that the ...
isnmr's user avatar
  • 41
45 votes
4 answers
5k views

Polynomial roots and convexity

A couple of years ago, I came up with the following question, to which I have no answer to this day. I have asked a few people about this, most of my teachers and some friends, but no one had ever ...
9 votes
5 answers
13k views

Get a point inside a polygon

I have a 2D polygon of arbitrary geometry. I need to find any point that is inside of that polygon. Taking the center won't work, because the polygon might not be convex. Is there a way to quickly ...
user10306's user avatar
  • 201
6 votes
1 answer
2k views

Continuous bijective way of representing a line on a plane

Is there a function $f(a,b)$ which maps ordered pairs to lines in a plane in a continuous, bijective manner? Here is the definition I am using for the limit with lines: a sequence of lines $L(1), L(...
yrudoy's user avatar
  • 435
18 votes
2 answers
916 views

Arrangements of points in the plane

Let $p_1,\ldots,p_n$ be a collection of distinct points in $\mathbb{R}^2$, no three of which lie on a line. For each $p_i$, let $\omega_i(p_1,\ldots,p_n)$ be the following ordered list (well-defined ...
Solid Snake's user avatar
3 votes
0 answers
153 views

The plane cut by grids

Suppose that one has an infinite two-dimensional regular grid of spacing one. When laid on the plane it cuts it into unit squares. Now take a second (identical) grid and place it with random shift ...
J.J. Green's user avatar
  • 2,497

1
6 7 8
9
10