All Questions
Tagged with computational-geometry algorithms
67 questions
11
votes
2
answers
3k
views
Algorithm for embedding a graph with metric constraints
Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, a poistive integer $d$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide ...
- 13.2k
4
votes
2
answers
1k
views
Polyline averaging
I'm trying to find a method that can take a collection of polylines, each described by a list of connected points on a plane, and find an "average" path through them. The input lines do not loop.
...
- 1
1
vote
0
answers
65
views
Algorithm to generate configurations with kissing number 12
That the kissing number of a sphere in dimension 3 is 12 is well known. However, it is also known that there is a lot of empty space between the 12 spheres. I deduce (am I wrong?) that there are many ...
- 3,065
4
votes
2
answers
219
views
Algorithm for grouping tetrahedra from Voronoi diagram
I have a set of 3D Voronoi generator points and their neighbouring points, which, when connected, should result in a Delaunay tetrahedralization. However, I'm having a hard time implementing this. My ...
- 701
10
votes
1
answer
3k
views
Computionally efficient vertex enumeration for (convex) polytopes
Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...
- 14k
3
votes
1
answer
271
views
Resultants and elimination theory
Consider an ideal $I = \langle f_1,\dotsc,f_n\rangle$ in the ring $k[x_1,\dotsc,x_m]$.
Define the $i$-th elimination ideal to be $I_i = I \cap k[x_{i+1},\dotsc,x_m]$.
For any two polynomials $f$ and $...
- 12.9k
5
votes
1
answer
255
views
Counting points above lines
Consider a set $P$ of $N$ points in the unit square and a set $L$ of $N$ non-vertical lines. Can we count the number of pairs $$\{(p,\ell)\in P\times L: p\; \text{lies above}\; \ell\}$$ in time $\...
- 121
5
votes
0
answers
167
views
Computing sums with linear conditions quickly
Let $f:\{1,\dotsc,N\}\to \mathbb{C}$, $\beta:\{1,\dotsc,N\}\to [0,1]$ be given by tables (or, what is basically the same, assume their values can be computed in constant time). For $0\leq \gamma_0\leq ...
- 20.2k
9
votes
3
answers
3k
views
Complexity of matching red and blue points in the plane.
I'm just asking because I'm curious.
I was seeking references on the following problem, that a friend exposed to me last holidays :
Problem
Given $n$ red points and $n$ blue points in the plane in ...
- 2,821
7
votes
1
answer
348
views
Finding maximal prefix of a simple curve
Let $S$ be a simple curve. I want to determine maximal prefix of $S$ contained in a unit circle. Is this possible, or has it perhaps already been solved in the past, and I am just unable to find an ...
- 71
1
vote
0
answers
38
views
Efficient solution to linear matrix equations
A general form for a linear matrix equation can be written as
$$AX + XB + \sum C_iXD_i$$
If $C_i$ and $D_i$ are all 0, then this simplifies into a well known and studied matrix equation that has an ...
- 11
20
votes
2
answers
25k
views
Partitioning a polygon into convex parts
I'm looking for an algorithm to partition any simple closed polygon into convex sub-polygons--preferably as few as possible.
I know almost nothing about this subject, so I've been searching on Google ...
- 4,713
15
votes
1
answer
359
views
Are hyperbolic $n$-manifolds recursively enumerable?
Fixing a dimension $n \ge 4$, is the class of closed hyperbolic $n$-manifolds recursively enumerable?
Since hyperbolic manifolds are triangulable I can reformulate this in the following more explicit ...
- 68.9k
5
votes
1
answer
169
views
Connectedness of semialgebraic sets via CAD
I do not know whether there is a standard or some traditional ways to decide whether a semialgebraic set is connected or not.
One way I know is the cylindrical algebraic decomposition (CAD) algorithm. ...
CommunityBot
- 1
9
votes
5
answers
13k
views
Get a point inside a polygon
I have a 2D polygon of arbitrary geometry. I need to find any point that is inside of that polygon. Taking the center won't work, because the polygon might not be convex. Is there a way to quickly ...
3
votes
0
answers
51
views
testing whether a polyhedral complex is convex
Definitions
A (polyhedral) cone in $\Bbb R^n$ is the solution set of a finite number of inequalities of the form $a_1x_1+\cdots+a_nx_n\geq 0$. Note that I don't require strict convexity, i.e. a cone $...
- 3,079
0
votes
1
answer
1k
views
Fast way to generate random points in 2D according to a density function
I'm looking for a fast way to generate random points in 2D according to a given 2D density function.
For instance something like this:
Right now I'm using a modified version of "Poisson disc&...
CommunityBot
- 1
4
votes
1
answer
204
views
Reference: Packing under translation is in NP
I am looking for a reference for a result that I am aware of.
Let me describe the result.
Given a polygon $C$ and polygons $p_1,\ldots,p_n$, it can be decided in NP
time, if $p_1,\ldots,p_n$ can be ...
CommunityBot
- 1
0
votes
0
answers
54
views
Attached convex "hulls"
Let $\mathcal{P}$ a finite set of points of a Euclidean $\mathbb{E}^n$ and take the union $\mathrm{U}(\mathcal{P})$ of all closed half-spaces defined by $n$ elements of $\mathcal{P}$ that contain only ...
- 13.2k
6
votes
1
answer
424
views
Probability of intersecting a rectangle with random straight lines
We are given a rectangle $R$ with sides lengths $r_1$ and $r_2$, contained in a square $S$, with sides lengths $s_1=s_2\ge r_1$ and $s_2=s_1\ge r_2$. $R$ and $S$ are axis-aligned in a cartesian plane $...
- 1,677
2
votes
0
answers
33
views
Algorithm for lightest unnested planar vertex-disjoint cycle-cover
Question:
given a finite set $\mathcal{P}$ of disjoint points in the Euclidean plane and the set $\mathcal{C}$ of all simple polygons whose corners are subsets of $\mathcal{P}$,
what is the ...
- 13.2k
3
votes
3
answers
2k
views
Is there a simple criterion to determine if two parallelograms intersect?
Assume we are given two parallelograms in the plane. How can I check if their intersection is nonempty?
Note that I do not need to actually find the intersection.
- 1
4
votes
2
answers
213
views
Algorithm for reporting all triangles with unique interior point
What is known about the complexity of and/or practical algorithms for reporting all triplets of points from finite set of at least four points of which no three are collinear in the Euclidean plane, ...
- 18.6k
5
votes
3
answers
149
views
Fast computation of a ball with radius r with largest number of input points
We are given a set S of n points equipped with some metric and an integer $r>0$. We define $B(x,r) \subseteq S$ (the ball with radius r centered in x) to be the set of points in S within distance r ...
- 6,476
2
votes
1
answer
213
views
Intersection of sphere with triangle containing moving vertices
First off, apologies if I cannot properly articulate my question in the most formal way. However, I believe my question should be simple enough to grasp anyhow.
In $\mathbb{R}^3$, I would like to ...
- 121
1
vote
0
answers
68
views
Projection of a polytope along 4 orthogonal axes
Consider the following problem:
Given an $\mathcal{H}$-polytope $P$ in $\mathbb{R}^d$ and $4$ orthogonal vectors $v_1, ..., v_4 \in \mathbb{R}^d$, compute the projection of $P$ to the subspace ...
- 11
4
votes
3
answers
2k
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. ...
- 111
3
votes
1
answer
589
views
How to partition a graph into N groups with M elements nearest?
This problem probably already here but I could not find the right words to find it.
I have a list with 1700 points (geographic coordinates) and a need to separate into 17 groups with 100 nearest. I ...
- 4,713
2
votes
1
answer
270
views
Algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane
I am trying to find an algorithm to compute the convex hull of a set of $m$ possibly intersecting convex polygons in the plane, with a total of $n$ vertices. Let $h$ denote the number of vertices on ...
7
votes
1
answer
3k
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 ...
- 151k
4
votes
2
answers
145
views
Complexity of Determining the Edges of Planar Convex Hulls
Question:
can the set of edges that resemble the convex hull ($CH$ for short) of $n$ points in the euclidean plane be determined in $O(n)$ time?
I know that the time complexity of determining the $CH$...
CommunityBot
- 1
4
votes
2
answers
734
views
Fast projection onto a subspace
Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{...
- 12.7k
2
votes
2
answers
687
views
Where to submit a new convex hull algorithm?
Recently, I devised a new convex hull algorithm. Is there any forum where I can submit my work?
- 4,713
1
vote
0
answers
117
views
The upper bound of the number of points of a convex hull formed by external co-tangents of circles
Consider the following problem: Given a rope to surround some circles, and minimize the length of the rope.
In order to solve the problem, we shall calculate all external co-tangents of these circles, ...
- 601
1
vote
0
answers
63
views
Congruency check for set of points in 3D using inertia tensor
You're given two set of points $A, B\subset \mathbb R^3:|A|=|B|=n$. You have to check if those sets are congruent, i.e. there exist some mapping $\sigma : A \to B$ and combination of translation and ...
- 1,171
13
votes
2
answers
664
views
Complexity of a weirdo two-dimensional sorting problem
Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...
- 1,083
0
votes
1
answer
67
views
Calculating a Measure of the Geometric Complexity of Planar Closed Polylines
Let $\lbrace p_1,\ \dots,\ p_n\rbrace$ be a set of points in the Euclidean plane and let $T_0 :=\left(p_1,\ \dots,\ p_n,p_1\right)$ be a Hamilton cycle through the set of points.
Question:
...
- 5,590
2
votes
1
answer
91
views
Generating Convex Polygonal Neighborhoods from Triangulations of Discrete Pointsets
The wellknown Delaunay Triangulation $DT$ has as a straight line dual the also wellknown Voronoi Diagram $VD$.
Both are most commonly defined in the Euclidean plane and are primarily beneficial for ...
CommunityBot
- 1
5
votes
1
answer
2k
views
Intersections of quadratic planes as elliptic curves
An elliptic curve defined over a field $k$ is a smooth projective curve of genus $1$, plus a $k$-rational point. Every elliptic curve can be written in a Weierstrass form, i.e. as a plane cubic curve ...
- 20.8k
21
votes
8
answers
4k
views
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 $r$...
- 151k
3
votes
2
answers
1k
views
Enclosing a set of ellipses within one ellipse
Is there an algorithm that takes in a set of ellipses and gives back an ellipse that encloses the original set of ellipses?
- 4,713
1
vote
1
answer
226
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
...
- 151k
6
votes
2
answers
2k
views
Find minimum-area ellipse which encloses two ellipses
I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can ...
- 151k
4
votes
1
answer
519
views
A brief question about the "Eight Queens" Puzzle
The classical Eight Queens puzzle asks whether it is possible to arrange $ 8 $ queens on an $ 8 \times 8 $ chess board, so that no two queens attack each other.
It is well-known that such ...
- 151k
2
votes
0
answers
486
views
Finding good high-dimensional sphere coverings in Euclidean space
Suppose we want to cover the unit sphere $\mathcal{S}^{d-1} := \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ with spherical caps $\mathcal{C}_{\mathbf{y}} := \{\mathbf{x} \in \mathcal{S}^{d-1}...
- 733
8
votes
3
answers
389
views
A simplified Art Gallery Problem in a matrix
Let's take a $m \times n$ matrix as an area with $m \times n$ blocks (likes a 2D-version of the world in Minecraft). We have to put some lamps in this matrix to illuminate the whole matrix. Here is ...
- 540
0
votes
0
answers
63
views
Distinguishing (possibly lower dimensional) $1$-skeleton of a regular graph inscribed in a sphere
Consider you have two (possibly same) convex $1$-skeleton of a regular graph $A$ and $B$ in $m$-dimensions inscribed in a sphere with possibly exponential number of vertices in $n$-dimension with ...
CommunityBot
- 1
2
votes
1
answer
69
views
Maximal opening angle of a polygon from a point [closed]
I'm looking for an algorithm that given a 2D convex polygon and point outside it, returns the two points of the polygon which are the two extremities of the polygon when viewed from that point.
One ...
- 151k
6
votes
1
answer
5k
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.
...
CommunityBot
- 1
4
votes
2
answers
2k
views
Breaking a rectangle into smaller rectangles with small diagonals
Say I am given a rectangle with dimensions $a \times b$ and an integer $n$. I'd like to break this rectangle into $n$ smaller rectangles $R_i$, and I'd like to make the maximum diagonal of any of ...
- 30.1k