**2**

votes

**1**answer

94 views

### mean length of the non-crossing graphs on n points

My original question is rather vague so I'll start with a precise example and then indicate possible generalisations.
Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...

**4**

votes

**1**answer

273 views

### “Average” Voronoi diagrams without probability?

A plane Poisson process with uniform intensity scatters "sites" about the plane. If I'm not mistaken, in a sense the "average" Voronoi diagram of that set of sites is a honeycomb. I know it's been ...

**3**

votes

**0**answers

39 views

### Find shift direction for min overlap area of 2 polygons

I have 2 arbitrary polygons (concave or convex) with certain overlap.
Now there is some relative shift between these 2 polygons (vector s with a constant length).
I want to find the direction of s ...

**3**

votes

**0**answers

141 views

### Dead Flies Problem [duplicate]

If a set of points in the plane contains one point in each convex region of
area 1, then can it have finite density?
what is the density of the points? In my understanding, it means the average ...

**3**

votes

**2**answers

218 views

### Combinatorial design for minimization problem over binary strings

Suppose the cost of a binary string $B$ of length $k$ is the number of $1$s that occur before the last $0$. For example, $1110$ has cost 3 while $0111$ has cost 0. Now suppose you can choose $k$ ...

**3**

votes

**1**answer

83 views

### Number of lattice polytopes contained in a given lattice polytope?

Given a (convex) lattice polytope, suppose we want to list or count all (convex) lattice polytopes (of the same dimension) contained in it. Are there efficient ways to do this?

**5**

votes

**3**answers

315 views

### How hard is it to determine if a weighted graph can be isometrically embedded in R^3?

Consider a graph $G$ with nonnegative edge weights.
Question: In $\mathbb{R}^3$, how hard is it to assign coordinates to vertices such that the Euclidean length of each edge is equal to its weight?
...

**5**

votes

**1**answer

154 views

### Given a polygon with holes, find a maximum distance pair in two subsets

I am curious about the following problem:
Given a polygon with holes and two convex subsets, $S$ and $T$, find points $s \in S, t \in T$ such that the shortest path between the two points has maximal ...

**5**

votes

**1**answer

191 views

### Simultaneous geometric separator

A geometric separator is a line that separates a given set of shapes to two subsets of approximately the same size (up to a constant), while intersecting only a small number of shapes. When a ...

**1**

vote

**1**answer

328 views

### Explicit computation of the action of a Dehn twist on the fundamental group of a surface

Let $S$ be a compact orientable surface of genus $g$. Now let $p\in S$ and $\gamma$ a closed simple curve on $S$ disjoint from $p$. It is not very difficult to compute the action of a Dehn twist along ...

**1**

vote

**2**answers

173 views

### A Claim on Typical Voronoi Cells

I am trying to prove the following claim (may be it has been proven).
Claim: Consider a set of points $\phi=\{x_1,x_2,...,x_i,...\}$ generated by a homogeneous PPP with rate $\lambda$ in the 2-D ...

**3**

votes

**2**answers

235 views

### Place N points in a 3d cube in a way that maximizes the minimum of their pairwise distances

Place $N$ points in a 3d cube in a way that maximizes the minimum of their pairwise distances.
The problem can easily be solved for $N\lt5$, but how to proceed for larger $N$?

**1**

vote

**0**answers

49 views

### Covering a set of points by bounded geometric object/objects

1) Let $S$ be a set of $n$ points in $R^d$. Now, given a bounded geometric object $G$, the problem is to check whether $S$ can be contained in $G$.
2) Also, in general setting, the problem is to ...

**6**

votes

**0**answers

82 views

### Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...

**1**

vote

**1**answer

69 views

### Smoothly deforming a set of three-dimensional points

I want to deform a 3D mesh according to 3 or more control points, meaning that the transformation is constituted by pre-images $c_i$ and images $c_i'$ of these control points. Each point of the mesh ...

**2**

votes

**2**answers

223 views

### Largest inscribed rectangle inside a convex polygon

It has been proved by Radziszewski in this paper
K. Radziszewski. Sur une probleme extremal relatif aux gures inscrites et circonscrites aux gures convexes. Ann. Univ. Mariae Curie-Sklodowska, Sect. ...

**4**

votes

**1**answer

129 views

### Maximal geometric mean of distances between points on an interval

Suppose I had T points in the interval $[0,1]$. Call them $e_1, \dots, e_T$.
Question 1:
What is a good nontrivial bound on the geometric mean of $$\{|e_i - e_j| : 1 \leq i < j \leq T \}, $$ as a ...

**3**

votes

**1**answer

170 views

### Hamiltonian circuit

Let us consider a disk with one labelled point on the boundary and $n$ labelled points in the interior.
Let T be a triangulation of the whole disk with vertices on the labelled points such that T ...

**3**

votes

**0**answers

324 views

### Intuition behind minimizing the Dirichlet energy of a mapping

What does minimizing the Dirichlet energy of a mapping $\Phi$ achieve intuitively?
Roughly it is the integral (or sum, if discrete) of $|\nabla \Phi(\;)|^2 dV$, with $V$ the volume.
So is it, in some ...

**4**

votes

**2**answers

229 views

### point in polytope

Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...

**4**

votes

**3**answers

656 views

### Intersection of Polyhedra

I'm writing a collision detection algorithm, and so far I've been using Joseph O'Rourke's book "Computational Geometry in C" as reference. It outlines an algorithm to determine whether a point is ...

**1**

vote

**2**answers

107 views

### Incremental structure of a delaunay triangulation

This would probably be considered a reference request, as I would imagine it has been studied extensively in earlier work. Say I have a collection of distinct points $X = \{x_1,\dots,x_n\}$ in the ...

**4**

votes

**1**answer

244 views

### Software computation with arithmetic schemes

For rings such as $\mathbb{Z}[x,y]$ is there software to compute any of:
1.) The integral closure of $\mathbb{Z}[x,y]/(f)$. de Jong has a very general algorithm that works in this context ...

**3**

votes

**2**answers

529 views

### Hyperrectangle partition of set of overlapping hyperrectangles

I have a set of $n$, $d$-dimensional hyperrectangles which may be overlapping in arbitrary ways. I would like to partition the area covered by this set into a set of non-overlapping hyperrectangles. ...

**4**

votes

**1**answer

566 views

### Finding the vertices of a convex polyhedron from a set of planes

I'm new to computational geometry and advanced mathematics in general here so bear with me. I've spent a decent amount of time attempting to figure out this problem and I just can't find a solution.
...

**3**

votes

**0**answers

82 views

### Computing with Graphs in Surfaces

I asked this question yesterday on math.stackexchange, but the only response so far hasn't really addressed the question, so I thought I'd cross-post it.
I am currently working on a research project ...

**17**

votes

**1**answer

575 views

### An NP-hard $n$ fold integral

We are given rational numbers $[c_1, c_2, \ldots, c_n]$ and $v$ from the interval $[0,1]$.
Consider the $n$-fold integral
$$
J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} ...

**4**

votes

**0**answers

184 views

### Upper bounds on art gallery problems using independent witnesses

Given a polygon $P$, the art gallery problem looks to find a smallest set of points that sees all of $P$. One way of bounding the number of guards necessary (from below) is to find a largest set of ...

**5**

votes

**1**answer

759 views

### Any reference to an algorithm for finding the largest empty circle on a sphere (with great-circle distance)?

Given a set $S$ of 2D points in the plane, there are known algorithms for finding the largest empty circle ($LEC$) of the set of points.
The $LEC$ problem is stated in this way: find a $LEC$ whose ...

**1**

vote

**1**answer

168 views

### Construction of an integral point set given the set of distances, its minimal description to get a measure of its complexity and its unique identifier

Given a set of distances between every pair of points of an integral point set $P$ of $n$ points; say $D = \{{d_i}\}$.
Q1. What is the least time complexity
possible/known for recreating the
...

**4**

votes

**1**answer

350 views

### Checking if a binary vector lies in the affine span of given binary vectors

Let $x_1,\ldots,x_N \in \{0,1\}^D$ be $N$ binary vectors in ${\mathbb R}^D$, assumed affinely independent. Is there an efficient algorithm for determining whether a new binary vector $x_{N+1}$ is in ...

**2**

votes

**1**answer

87 views

### Covering the annulus of d-cube

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c+x \mid c\in C \}$, for some $x\in R^d$, is called a homothetic copy of $C$. The ...

**0**

votes

**0**answers

40 views

### Covering the annulus of symmetric convex body

Consider a symmetric convex body $A$ in $R^d$. Now, we draw another object, $A'$, concentric and translated with respect to A and having radius slightly greater than twice to the radius of $A$.
Now ...

**4**

votes

**1**answer

410 views

### Algorithms for covering a rectilinear polygon using the same multiple rectangles

Sorry for the crossing-posting: original post is here
All angles of the polygon (representing a room) are right. It may be convex or concave. Use rectangles of the same size (representing a sensor ...

**4**

votes

**1**answer

221 views

### convex polyhedron in the unit cube

Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $\operatorname{conv}(Q) \supseteq P$ where ...

**4**

votes

**1**answer

327 views

### Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...

**3**

votes

**1**answer

144 views

### Number of isomorphism classes of triangulations of a convex polygon

The number of triangulations of a convex $n$-gon is $C_{n-2}$ the $n-2$nd Catalan number. What I am wondering, is if there is a way to enumerate the isomorphism types of these as graphs? I am ...

**2**

votes

**1**answer

347 views

### Regularity of Delaunay triangulation of a hypercube

First using a three dimensional unit cube as an example for the term "regularity", we can have two possible triangulations:
(A)
(B)
We say the lower triangulation is more "regular" than upper ...

**2**

votes

**2**answers

1k views

### Trilateration problem

When trying to develop an algorithm for a program, I got with the following problem:
Determine the approximate location of $O$, if you can take finite samples $P_n$ from known locations and always ...

**47**

votes

**2**answers

1k views

### Reasons to prefer one large prime over another to approximate characteristic zero

Background:
In running algebraic geometry computations using software such as Macaulay2, it is often easier and faster to work over $\mathbb F_p = \mathbb Z / p\mathbb Z$ for a large prime $p$, rather ...

**0**

votes

**2**answers

257 views

### Higher dimensional convex hull

Let $CH(S)$ be a convex hull of a finite set $S$ and denote the set of all the vertices of $CH(S)$ as $Vert(S)$. For a vertex $v \in Vert(S)$, it has an associated set $E(v)$ which is defined as ...

**1**

vote

**1**answer

80 views

### Inferring the properties of a visibility blocker tangential to a point-like light source

Imagine there's a point-like particle undergoing radioactive decay at some position $(0,0,0)$ in Euclidean $3$-space. We encapsulate this particle with a spherical detector for the decay products it ...

**1**

vote

**0**answers

100 views

### Boundary surfaces in a 3d Voronoi tessellation with obstacles

Let $x_1,\dots,x_n$ be a set of points in $\mathbb{R}^3$ and let $\mathcal{O}_1 ,\dots, \mathcal{O}_m$ denote a set of polyhedral obstacles. What is the name for the surfaces that describe the ...

**6**

votes

**2**answers

347 views

### Intersection of 2 visibility polygons

Let $P$ be a simple, closed and bounded polygon and $p_1,p_2 \in \mathrm{int}(P)$ be two points in its interior. Is it true that the intersection of the visibility polygons of $p_1$ and $p_2$ is ...

**3**

votes

**1**answer

133 views

### The discrete theory of compressible fluids dynamics

I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete exterior calculus, ...

**0**

votes

**2**answers

178 views

### Incidences of quadratic forms and points

Is there anything that is known about what is the maximal number of incidences between quadratic forms and points? I looked at the internet and I haven't found anything that works for something that ...

**1**

vote

**0**answers

107 views

### minimizing the sum of euclidean norms

minimizing the sum of euclidean norms with box constraints
I am a graduate student in computer science, making a thesis on uncertainty geometry. During my thesis I came across the following ...

**3**

votes

**2**answers

328 views

### Questions on Discrete Exterior Calculus in numerial computing

I have several questions about the Discrete Exterior Calculus (DEC) in the numerical method for solving partial differential equation in physics:
(Discrete Exterious Calculus is the newly developed ...

**1**

vote

**1**answer

61 views

### Uniformly sampling the solution space for points where the free termini of two rays, anchored at 3-space points, can intersect

I have two rays, one of length $L_1$ and one of length $L_2$. I anchor these rays, each at one end, on the 3-space points $p_1$ and $p_2$. Assuming that the Euclidean distance between $p_1$ and ...

**3**

votes

**1**answer

164 views

### Triangulation of the surface determined by sampling two of its cross-sections

I have a data set that essentially looks like the picture below, i.e., it's given by sets of points in $\mathbb{R}^3$ that sample the cross-sections of a certain surface that in principle I do not ...