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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6
votes
3answers
2k views

Random Sampling a linearly constrained region in n-dimensions…

Hi, So here is my problem: Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$ $x_n \le c_n$ and $\sum_{n=1}^N x_n = 1$ find an ...
7
votes
2answers
1k views

Conic hulls and cones

Suppose I have a number of vectors in $\mathbb{R}^n.$ The first question is: what is the most efficient algorithm to compute their "conic hull" (the minimal convex cone which contains them)? The next ...
6
votes
1answer
834 views

Algorithm for embedding a graph with metric constraints

Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide if there is an ...
14
votes
1answer
341 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 ...
5
votes
0answers
202 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 ...
2
votes
1answer
176 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. $N$) ...
3
votes
1answer
4k views

split polygon into minimum amount of rectangles and triangles

Hi is there an algorithm which cuts a polygon into a minimum amount of preferably rectangles and where not possible (e.g. edges) into triangles?
8
votes
2answers
702 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 ...
13
votes
2answers
3k 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 $X$. ...
6
votes
1answer
139 views

Angle subtended by the shortest segment that bisects the area of a convex polygon

Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could ...
5
votes
0answers
140 views

Euclidean Minimum Spanning Trees Restricted to One Vertex Per Grid Cell

Given an $n \times n$ grid with unit grid cells, and one point from the interior of each cell, what is are best possible lower and upper bounds for lengths of minimum spanning trees? The lower bound ...
5
votes
3answers
334 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? ...
2
votes
0answers
94 views

The intersection of two $l_1$ balls

Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ with respect to the $l_1$ norm that have different radii and different centers. Is there an upper bound for the number of vertices that $B_1\cap ...
4
votes
2answers
381 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 ...
4
votes
2answers
146 views

Construct polygon/polyhedron containing all points not externally visible w.r.t given polygon/polyhedron?

Is there an algorithm to construct a polyhedron containing all points in space for which there exists no ray to infinite not intersecting a given polyhedron? In 2D, we could consider polygons. For ...
4
votes
2answers
245 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 ...
3
votes
1answer
2k 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 ...
3
votes
3answers
6k 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 ...
2
votes
1answer
88 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 ...