**6**

votes

**0**answers

78 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

67 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

208 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

128 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

169 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

282 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

222 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

583 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

106 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

241 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

492 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. ...

**3**

votes

**1**answer

419 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

76 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

550 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

177 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

692 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 ...

**4**

votes

**1**answer

337 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

85 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

37 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

366 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

213 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

314 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

135 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

304 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 ...

**46**

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

248 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

96 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

342 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

129 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

175 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

97 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

308 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

56 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

148 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 ...

**7**

votes

**0**answers

141 views

### Fractional Helly for more than one piercing

Fractional Helly Theorem says the following:
For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...

**3**

votes

**1**answer

204 views

### What properties does generalized Delaunay triangulation have?

Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...

**2**

votes

**3**answers

2k views

### Intersection of Cones in Three Space

In several branches of applied mathematics the problem arises to describe the intersection of two cones in three space.
I have searched and found a few references that discuss the problem for cones ...

**1**

vote

**2**answers

835 views

### Projection of a point to a convex hull in d dimensions

Hi,
I've got n points in d dimensions (typically n is around 30k-60k and d is 5 or 6). I'm using qhull to calculate the Delaunay triangulation and the convex hull of the set of points.
You can ...

**3**

votes

**2**answers

525 views

### Find longest segment through centroid of 2D convex polygon?

Given a 2D convex polygon P and its centroid C, how do I find the longest line segment passing through C, where the endpoints of the segment lie on the boundary of P?
Intuitively I imagine there ...

**3**

votes

**1**answer

1k views

### Subtract Rectangle from Polygon

I'm looking for an algorithm that will subtract a rectangle from a simple, concave polygon and return a remainder of polygons. If the rectangle encloses the polygon, the remainder is null. In most ...

**0**

votes

**0**answers

223 views

### Determining the simplices in freudenthal triangulation

HI,
I have a doubt on Freudenthal Triangulation. I want to partition a simplex into finer simplices.
The FT gives me the vertices of the simplices which partition my original simplex into ...

**1**

vote

**0**answers

352 views

### Solving 3D equation system (inverse-projecting a triangle)

Please, how is the equation system below named exactly (to search further literature)?
Does it have an analytical solution? If it doesn't, then what could be the fastest numerical method for it ...

**1**

vote

**0**answers

156 views

### Compute generalized pentagram map

Hi,
(This is my first question on MathOverflow! :-)
Imagine you have a set of points $S = \{p_1, \ldots, p_n\}$ in $\mathbb{R}^d$, of which $t$ are "bad". I want to compute a "safe convex hull", ...

**10**

votes

**3**answers

485 views

### Algorithms in Invariant Theory

Let $V$ be a polynomial representation of the general linear group $\Gamma:=\DeclareMathOperator{\Gl}{Gl}\Gl_n(\newcommand{\C}{\mathbb C}\C)$.
In chapter 4.6 of his book "Algorithms in Invariant ...

**5**

votes

**5**answers

584 views

### Finding an axis-aligned ellipsoid of minimal volume which contains a given ellipsoid

A friend asked me to post the following question. He's not an MO user and felt it would be better received if asked by someone who was already known to the community. This is not my area, but I'll do ...

**7**

votes

**1**answer

2k views

### Get Largest Inscribed Rectangle of a Concave Polygon

I'm looking for an algorithm to find a set of largest inscribed rectangles of a concave polygon where each rectangle must be collinear with one of the edges of the polygon.
In other words, I want to ...

**8**

votes

**2**answers

2k views

### Covering a Polygon with Rectangles

I am tyring to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle.
I thought about ...

**3**

votes

**2**answers

425 views

### The (Sigma) Algebra of Convex Sets

This is a question-by-proxy for a colleague from computer science. I'm sure many people here are already aware that convex decomposition forms an important sub-field of both computational geometry and ...

**2**

votes

**2**answers

592 views

### Decomposing a polygon with holes

It is known that given a polygon $P$ with holes it is NP-hard to find a decomposition of $P$ into convex polygons, s.t. their number is minimized (even if Steiner points are allowed).
I am wondering ...