Using computers to solve geometric problems. Questions with this tag should typically have at least one other tag indicating what sort of geometry is involved, such as ag.algebraic-geometry or mg.metric-geometry.

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Complexity of Untwisting Polygons

What is the complexity of the following task: given a sequence $p_1, ..., p_n, p_1$ that defines a closed polyline in the euclidean plane, what is the complexity of finding a reordering of the points, ...
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1answer
173 views

Ways to show a system of polynomial equations has no solution

I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$: $$ \begin{cases} 2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\ ...
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2answers
270 views

Random Vornoi Diagrams (particular measures)

This is my second question about Random Voronoi diagrams, in my first question was given some excellent advice but i was not clear in explaining what i was looking for. I'm interested to know ...
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596 views

Random Voronoi Diagrams

I'm interested in what research has already been done with regards to the statistics of random voronoi diagrams. I have had a look on google scholar and results are a little inconclusive. I'm ...
2
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1answer
87 views

Computable link invariants

I am interested in the following situation: given a braid $B$, it induces a link $L$ in a pretty straightforward way ("glue" the endpoints, like here). For a braid $B$, we know how to represent it in ...
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1answer
220 views

Hilbert function of points in $\mathrm{P}^2$

Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If $$ ...
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2answers
2k views

Weighted area of a Voronoi cell

Let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w_1,\dots,w_n\}$ denote a set of weights corresponding to the $n$ points in $X$. ...
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0answers
32 views

Any software that can symmetrize input sets?

Is there any software that contains symmetrization techniques ex. polarization, Steiner Symmetrization etc. I suppose not. Which software would you suggest for rigid transformations? Thank you
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164 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 ...
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1answer
60 views

Finding a minimum covering of a polygon with interesting shapes

After reading many papers about problems of minimum polygon covering, I found out that there are four different types of units that are considered for covering polygons, in increasing order of ...
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1answer
55 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 ...
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1answer
219 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 ...
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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 ...
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2answers
961 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 ...
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2answers
177 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 $k$ ...
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0answers
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plotting parametrized algebraic curves near singularities

I have a parametrized algebraic curve: x(t)=A(t)/D(t); y(t)=B(t)/D(t); with A(t) and B(t) being polynomials in t. The curve is solution of a linear system in two unknowns x and y with Cramer's ...
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132 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 ...
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1answer
69 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?
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234 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 ...
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9answers
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Determine if circle is covered by some set of other circles

Suppose you have a set of circles $\mathcal{C} = \{ C_1, \ldots, C_n \}$ each with a fixed radius $r$ but varying centre coordinates. Next, you are given a new circle $C_{n+1}$ with the same radius ...
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3answers
267 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? ...
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1answer
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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 ...
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3answers
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 ...
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2answers
1k views

Anuloid (Torus) x line intersection

Hi, I need calculate ray (line) intersection with torus for my ray-tracing program (I know, its to graphics, but i need math behind it). I can solve equation of order x^4, but thats too way slow ...
5
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1answer
121 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 ...
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1answer
471 views

Graph drawing: unrooted undirected tree graphs with specified edge lengths.

Has Joseph Felsenstein's equal daylight layout been analyzed by the graph drawing community? The following description is taken from his drawtree documentation: "This iteratively improves an ...
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1answer
202 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 ...
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2answers
157 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 ...
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3answers
429 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 ...
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2answers
193 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 $N$?
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76 views

minimizing the sum of euclidean norms with box constraints

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 ...
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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 ...
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0answers
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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 ...
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2answers
206 views

Which data structure should I use for hierarchical T-meshes and PHT-splines?

Recently, I'm working on something about polynomial splines over hierarchical T-meshes, which is basically a rectangular grid that allows T-junctions. I want to do some numerical experiments but I ...
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0answers
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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 ...
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1answer
61 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 ...
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2answers
159 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. ...
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1answer
123 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 ...
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4answers
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Computational software in Algebraic Topology?

I was wondering if there is any good software out there that allows you to do specific computations in algebraic topology. For example: Create a simplicial complex/set and ask questions about its ...
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0answers
207 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 ...
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1answer
239 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 ...
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2answers
199 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 ...
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1answer
280 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 ...
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3answers
2k views

n-dimensional voronoi diagram

Hi, I need to compute the voronoi diagram of a set of points in $R^n$. I'm quite unschooled on the topic, could someone point me to the right references so that I can a) understand the theory behind ...
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2answers
104 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 ...
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2answers
383 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. ...
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1answer
256 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. ...
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3answers
424 views

How to find the minimum number of hyperplanes to define a convex hull?

I have the following problem: I have a convex hull $\Omega$ defined by a set of n-dimensional hyperplanes $S = [(n_1,d_1), (n_2,d_2),...,(n_k,d_k)]$ such that a point $p \in \Omega$ if $n_i^T p \geq ...
3
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1answer
113 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 ...
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0answers
70 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 ...