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3
votes
0answers
73 views

Moving circular disk between two parallel sinusoidal curves [migrated]

Find the largest radius of the circle that can be "rolled" between the curves $y = sin(x)$ and $y = sin(x)+1$. After two weeks of research, I finally give up.
7
votes
1answer
103 views

Elementary proof of a triangular grid lemma

I am looking for an elementary proof of the following lemma, which concerns what Green and Tao call "triangular grids" (see arXiv:1208.4714). Let $a_1$, $a_2$, $a_3$, $a_4$, $b_1$, $b_2$ be six ...
5
votes
1answer
77 views

Penrose tiling substitution is bijective

Let $\mathcal{P}$ a Penrose tiling built by a substitution $\omega$ with two triangles. It is claimed, for instance, in the article of Anderson and Putnam "Topological invariants for substitution ...
1
vote
1answer
306 views

Explicitly describing the region of the plane “outward of” a simple, open, oriented, cubic curve $c:(0,1)\to\mathbb{R}^2$

Some Context: I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an ...
1
vote
0answers
39 views

Projecting a convex partition onto a convex set

Say that $X$ and $Y$ are two convex regions in the plane, and suppose that $X \subset Y$. Further suppose that $Y$ is partitioned into disjoint convex subsets $Y_1 ,\dots, Y_n$. Is there a way of ...
16
votes
1answer
359 views

What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?

As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...
2
votes
1answer
118 views

What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?

Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...
10
votes
0answers
135 views

Dividing a convex region to minimize average distances

Let $C$ be a convex region in the plane with area 1 that contains distinct points $p_1,\dots,p_n$. Say I'd like to divide $C$ into $n$ pieces $C_1,\dots,C_n$, each of area $1/n$, and I'd like to ...
4
votes
1answer
121 views

Segments on a family of parallel lines

Let $\{l_i:i\in I\}$ be a family of parallel lines on the plane $\mathbb{R}^2$. Suppose for each $i\in I$ there is a closed segment $s_i\subset l_i$. Moreover, for each triple $i_1,i_2,i_3$ there ...
0
votes
0answers
110 views

How large can a set of nearly equidistant points be?

Suppose that $D$ is a set of points in $\mathbb{R}^{k}$ such that all pairwise distances between them belong to $[1,1+\epsilon]$. It seems that such a set cannot be very large and that its ...
0
votes
0answers
30 views

Is there a term for “ranked distance” matrices?

In a n by n "ranked distance matrix" each element has a rank $r_{ij}$ between 1 and n that indicates it is the $r_{ij}$th smallest element in column $i$ of a corresponding Euclidean distance matrix. ...
6
votes
1answer
129 views

Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...
6
votes
1answer
166 views

Did Lucas discover Lucas circles?

MathWord's article on Lucas circles traces the name to a little-known 1973 publication. These interesting circles have found their way into several 21st century publications, including the online ...
7
votes
0answers
124 views

Ricocheting pinball-like shot: Complexity?

Suppose one has $n$ perfect two-sided mirror segments in the plane $\mathbb{R}^2$. The segments are open, excluding their endpoints. They are disjoint as closed segments, i.e., no pair shares an ...
4
votes
1answer
85 views

Geometric realization of an abstract triangulation of the plane

Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy ...
2
votes
0answers
110 views

Examples of Geometric Constructions in Higher Dimensions

The classical problem of geometric construction seems to be restricted to planar Euclidean Geometry with straight edge and compass as the only admissible "construction-tools". I would like to ...
4
votes
1answer
529 views

Focus of parabola using only a ruler

It is an easy exercise that using ruler and compass one find the focus of a given parabola. Can one do the same using only a ruler? -- if not, why?
2
votes
1answer
85 views

Measuring the Randomness and Statistics of Convex Polygons

How can I tell, how likely it is, that a given convex polygon with a sufficiently high number of edges is random and, if so, what kind of randomness it is (e.g. white noise)? What is known about ...
10
votes
5answers
500 views

ten concurrent lines

this relates to a question asked at MSE. i was able to find an answer using complex numbers. here is the question: there are five points on a circle. take any three points, through the centroid of ...
9
votes
0answers
282 views

How can we join two points with a small ruler? [closed]

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this? Imagine a method that works when $AB$ is really huge and ...
8
votes
1answer
707 views

How large can you draw an island on a map?

A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...
2
votes
1answer
76 views

Calculating the “Belvedere Hull” of a Simple Planar Polygon

As an informal motivation the problem, imagine a tower with polygonal footprint, that is located in a beautiful landscape, the "Belvedere Hull" is then related to the directions, in which one would ...
28
votes
2answers
678 views

Term for “uncheckable constructions”

Is there a term for "uncheckable geometric constructions"? Say, Angle Trisection and Doubling the Cube are checkable; i.e., if the answer is given one can do finite Compass-and-straightedge ...
3
votes
1answer
129 views

Three-dimensional Apollonian spirals

Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$. Let ...
6
votes
4answers
211 views

Inside-out polygonal dissections

A dissection of a polygon $P$ is a partition of $P$ into a finite number of pieces, which can then be rearranged (via planar translations and rotations) and joined (without overlap) to form a new ...
6
votes
2answers
334 views

Are angles between points enough to decide the realizability?

Let n points in the plane be given whose coordinates we don't know. Assume, however, that for any triple of the points we know the angle. Question: Can we decide whether the n points are realizable ...
0
votes
1answer
63 views

Existence of half-planes with respect to regular open sets of the Euclidean plane

I initially asked this question at math.stackexchange.com but there was no reaction, so I thought this may be a good idea to transfer it to mathoverflow.net Let ...
9
votes
1answer
275 views

Generalized Hlawka inequality

Let $E$ be a vector space over the real (the complex case is interesting too). We consider functions $f:E\rightarrow\mathbb R$ which satisfy the homogeneity property $$f(\lambda x)=|\lambda|\,f(x).$$ ...
3
votes
1answer
204 views

Random non-intersecting circles in the plane

If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...
3
votes
0answers
83 views

Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...
5
votes
1answer
221 views

Maximum number of general-position points with mutual rational distances?

Richard Guy has shown that there are six points in the plane—no three collinear, no four cocircular—such that all interpoint distances are rational. Guy, Richard. Unsolved Problems in ...
8
votes
1answer
254 views

Dubins car shortest paths: Decidable?

A Dubins car follows a Dubins path in $\mathbb{R}^2$, with constant wheel speed and limited turning radius. It is known that the shortest Dubins path in the absence of obstacles follows circular arcs ...
6
votes
4answers
364 views

What can be said of the structure of a metric space without isosceles triangles?

This is a question that has been bothering me in the back of my head for quite some time. Suppose we have a metric space $X$ with metric $\mathrm{d}$. By an isosceles triangle we mean a tuple of ...
2
votes
2answers
114 views

Maximum possible number of similar three-colored triangles

I want to maximize the number of similar triangles with vertices from three fixed sets, one vertex from each set. For example, if you fix two points $X$, $Y$ (i.e. two sets with only one member), then ...
-3
votes
2answers
147 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 ...
2
votes
1answer
236 views

Tools for Removing Radicals from Equations

I am currently doing some investigations on Sylvester's 4 Point Problem Probability of 4 Points being in Convex Configuration and repeatedly face the problem of solving equations between sums of ...
1
vote
0answers
123 views

A conjecture about cross sections of a pyramid [closed]

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. This is a conjecture I came across while trying to solve this problem. I was ...
2
votes
0answers
181 views

Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$): $\min_{x\in U}\|x-y\|$. I believe the minimizer $x_{U}^{*}$ is ...
3
votes
1answer
307 views

Conditions for a parametric curve to avoid self-intersection?

Suppose a planar curve $C$ is defined by parametric equations in $t$: $x(t)$ and $y(t)$. Are there conditions on these two functions that guarantee that $C$ does not self-intersect? For example, the ...
2
votes
0answers
152 views

Probability of 4 Points being in Convex Configuration

Background of my question is, that I would like to implement a parallel preprocessing for a constructing the convex hull of very huge number of points in the euclidean plane; the idea is to process ...
4
votes
1answer
433 views

Reference request: a differential equation in elementary geometry

15 hours and four up-votes but not a word from anybody. That's the result of this question to stackexchange. My question is where the following differential equation arises naturally and where it ...
3
votes
2answers
324 views

Points contained in a disk [closed]

I have a question, but not sure how to prove this. We are given $n$ points in the Euclidean plane such that there exists no disk of radius $a$ which contains all of the points. Conjecture: There ...
7
votes
2answers
223 views

How to find a tetrahedron that covers four points?

I’m looking for an explicit formula for the vertices of a regular tetrahedron that covers four given points. In particular: Given four distinct real numbers $a_1$, $a_2$, $a_3$, $a_4$, is there a ...
4
votes
1answer
461 views

What is the correct preposition? (And is there one?)

I just stumbled upon a linguistic problem I wasn't able to resolve via web search. Suppose we're given some geometric set $A$ and subset $B\subset A$. Isn't there a compact way of saying that there ...
2
votes
1answer
219 views

Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes. For a given $\delta$, let $n_\delta$ be the number of faces ...
3
votes
0answers
42 views

Number of not self-intersecting closed paths spanning $n$ iid uniform points

Let $X_1,X_2,\dots,X_n$ be independent uniform variables in the square. What is the number of piece-wise linear paths which vertices are all the $X_i$ and that do not self-intersect? In other words, ...
1
vote
2answers
99 views

Successive Inner or Outer Approximation of Simple Polygons with Hierarchies of Implicit Functions

The problem I want to solve, is to quickly decide, whether a point $p=(x^*,y^*)$ is inside or outside of a polygon $P := (p_1, p_2,..., p_n=p_1), p_i := (x_i,y_i)$, with $n$ potentially very large. ...
1
vote
3answers
527 views

Characterization of Angles Trisectable with Straightedge and Compass

Lindemann's prove of the transcendence of $\pi$ has settled the question, whether an arbitrary angle can be trisected, using straightedge and compass alone, to the negative. In the following, ...
3
votes
1answer
86 views

Number of small projections

Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of ...
10
votes
3answers
419 views

The intersection of $n$ cylinders in $3$-dimensional space

A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...