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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2answers
274 views
algebraic topology and 3d/4d printing [closed]
I googled for papers on applying algebraic topology to 3d/4d printing. It just seems to me that there has to be a connection. Any help, kind audience?
edit: 4d printing means 1-parameter families of ...
1
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0answers
22 views
Determining feasibility of specific intersection structures between closed paths in the plane
I have two arrays, each containing a different ordering of the same set of integers. Each integer is a label for a point in which two closed paths intersect in the plane. The two arrays are ...
2
votes
1answer
56 views
What is Known about Preprocessing for Stabbing Queries?
In a concrete setting, I have the following problem:
given a fixed set of simple objects (e.g. disks or, convex polygons with few vertices), I need to quickly report the objects that are hit (i.e. ...
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votes
1answer
78 views
Practical Algorithm for Comparing the Discrepancy of Point Sets (on Unit Hyper Spheres)
I have devised a simple geometric algorithm for generating a sequence of points on unit hyper spheres; that algorithm depends on a single real parameter, which I would like to optimize in order to get ...
2
votes
1answer
206 views
Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?
Not sure whether this question belongs here or math.stackexchange.
You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...
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vote
2answers
194 views
Regular paths along surface of sphere
I'm trying to create a program where a small ball is supposed to move along the surface of a sphere, which is given by its radius $r$ and the center $c$.
The movement should be repetitive, so that ...
11
votes
1answer
281 views
Are all well behaved “mean” functions on $\mathbb{R}^+$ equivalent?
Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
...
7
votes
0answers
121 views
Nearest Point to a real algebraic set
Suppose I have a compact bounded real algebraic (eventually: or analytic or semialgebraic or semianalytic set) $V$ in $\mathbb R^3$ and a point $x\in\mathbb R^3$ not in $V$. How much do we know about ...
2
votes
1answer
62 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 ...
1
vote
1answer
133 views
NP hard problems on UD graphs
I'm reading up on NP hard problems in Unit Disk graphs. I'd like to point out i'm fairly new to this NP hard stuff so i'm trying to get around how to prove something is NP hard.
...
1
vote
0answers
44 views
Representing a Pullback as an Infinite Matrix
Let $M$ and $N$ be manifolds and let $T: M \to N$ be a bijective map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) be the space of all functions from $M$ (resp. $N$) to ...
2
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1answer
138 views
Inverse Problem for Pullback
Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) ...
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1answer
123 views
Exploiting the Linearity of the Pullback [closed]
Edit: This question has been significantly revised.
Some recent developments in computational geometry (for example see http://geometry.stanford.edu//papers/fmfrmbs-obsbg-12/fmfrmbs-obsbg-12.pdf) ...
0
votes
0answers
90 views
Trilateration issues, when circles don't intersect
I'm working on Indoor localization where I've deployed multiple iBeacons in my environment. I'm taking distances from all the beacons through their RSSI values. They are not 100% accurate though. Now ...
0
votes
2answers
202 views
Determine the boundary points of a set of points [closed]
I have a set of points $S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$. Then how to find the boundary points (which is a subset of $S$) of $S$?
There are methods like convex hull, concave hull and ...
3
votes
2answers
74 views
Average vertex degree in finite Delaunay triangulations in high dimensions
In $\mathbb{R}^2$ it's known that with a "random" point configuration, the average degree of a vertex in its Delaunay triangulation is 6.
Does anyone know of a similar result in higher dimension? I ...
1
vote
0answers
42 views
Non-Convex Polygons with “Antipodal Visibility”
by "antipodal visibility" of planar, simple polygons I mean the following property:
if two points $p$ and $q$ on the polygon's boundary divide the polygon's boundary into two polylines of equal ...
0
votes
1answer
265 views
Is there a Gröbner basis analogue that exists for vector spaces?
Suppose I have a coordinate system $t_1,\ldots t_N$ with a lexicographical ordering. Let LT denote choosing the lowest term of a polynomial with respect to this ordering. e.g. LT$(t_1 + t_2)=t_2$.
...
2
votes
0answers
171 views
Dissection of a polygon into convex polygons
Problem: for a fixed integer $m\geqslant 3$ find all $n$ such that no $n$-gon can be dissected into convex $m$-gons.
I would be very grateful for any information on this problem.
Remark 1. There ...
8
votes
1answer
217 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
2answers
119 views
Number of edges in linklessly embeddable graphs
Consider graphs over $n$ nodes.
What is the maximum number of edges of a linklessly embeddable graph?
A more general question is the following. Given $\mu$ what is the maximum number of edges of ...
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votes
1answer
70 views
Paper on unit disk graphs
I was wondering if anybody knows of a 'link' to the paper by Marathe 1995 et al on analysis of the greedy algorithm for finding a Max independent set in Unit Disk Graphs?
2
votes
2answers
87 views
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, ...
10
votes
1answer
612 views
Ways to show a system of polynomial equations has no solution
I came across the following system of polynomial equations on $X_1,\dots,X_{m-2}$:
$$
\begin{cases}
2X_{2s}+\sum\limits_{t=1}^{2s-1}(-1)^tX_tX_{2s-t}=0,\quad s=1,\dots,\frac{m}{2}-1,\\
...
5
votes
2answers
288 views
Random Vornoi Diagrams (particular measures)
This is my second question about Random Voronoi diagrams, in my first question was given some excellent advice but i was not clear in explaining what i was looking for.
I'm interested to know ...
8
votes
2answers
640 views
Random Voronoi Diagrams
I'm interested in what research has already been done with regards to the statistics of random voronoi diagrams. I have had a look on google scholar and results are a little inconclusive. I'm ...
2
votes
1answer
104 views
Computable link invariants
I am interested in the following situation: given a braid $B$, it induces a link $L$ in a pretty straightforward way ("glue" the endpoints, like here). For a braid $B$, we know how to represent it in ...
2
votes
1answer
233 views
Hilbert function of points in $\mathrm{P}^2$
Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If
$$
...
0
votes
0answers
53 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
3
votes
1answer
82 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 ...
2
votes
1answer
88 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
1answer
237 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
0answers
38 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 ...
0
votes
0answers
58 views
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 ...
3
votes
0answers
134 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
2answers
198 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
1answer
75 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
3answers
298 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
1answer
133 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
1answer
186 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
1answer
280 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
2answers
164 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
2answers
213 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
0answers
47 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
0answers
77 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
1answer
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
2answers
194 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
1answer
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
1answer
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
0answers
261 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 ...
