# Tagged Questions

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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### 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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### 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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### 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 $r$...
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### Missing document request

I received a request for another long-lost document: I am wondering if there is any way I might obtain a copy of The geometry of circles: Voronoi diagrams, Moebius transformations, ...
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### 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 ...
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### Is every graph an edge-crossing graph?

Consider a circular drawing of a simple (in particular, loopless) graph $G$ in which edges are drawn as straight lines inside the circle. The crossing graph for such a drawing is the simple graph ...
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### 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 ...
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### Degree of generators of irreducible components

Let $V$ be a Zariski-closed subset of $\mathbb{A}^n_k$, where $k$ is an algebraically closed field. Assume that $V$ may be defined by polynomials of degree at most $d$ (or to put it otherwise $V$ is ...
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### Can taking the projective closure of an affine variety increase the degrees of its ideal generators?

Say we have some equations $f_1(x)=0, \ldots f_k(x)=0$ defining a variety $X$ in ${\mathbb C}^n$ (not necessarily a minimal number of generators, and not necessarily of minimal degree), and suppose we ...
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### 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 ...
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### 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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### Computational methods for dealing with geometrically complicated solid boundaries in fluid-air interface problems

Hello, I am a PhD student who does not have extensive computational experience seeking advice from those experienced with computational modelling as to which method would be most appropriate for ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Shortest Path in Plane

Hi I thought about the following problem: Given a polygonal subdivision of the euclidian plane where each of the polygons has a speed associated with it, and given two points s,t, I'm interested in ...
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### Minimum separating subdivision in Plane

Hi I was thinking about the following problem: Given a planar Graph embedded in the plane and a set of points $P$ contained in the faces (no face contains more than one point) I want to determine ...
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### Algorithm for k-medians in a convex polygon

Are there any known approximation algorithms or exact solution schemes for the k-medians problem in a convex polygon? That is, placing a collection of points $p_1,\dots,p_k \subset \mathbb{R}^2$ in a ...
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### 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 ...
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### 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 ...
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### Do computational geometers use Lagrange multipliers?

Can anyone point me to an example of a problem that (more or less) originated in computational geometry whose solution requires the use of Lagrange multipliers (or Kuhn-Tucker conditions, or dual ...
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### How to compute the average distance till intersection within a triangle in R^2?

Lots of simple questions because I am a noob. You are given 3 points in R^2; A, B, C forming a triangle with area > 0. You pick an arbitrary point inside ABC and an arbitrary direction. After some ...
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### Algorithm that generates a n-simplex that cover n-polytope?

Given an $n$-cube with unit volume, is there any algorithm that generates a $n$-simplex that covers the $n$-polytope?
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### 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 ...
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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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### Segments of Voronoi Diagrams on smooth manifolds. Are they geodesics?

Let $S$ be a patch of a smooth 2-manifold in $\mathbb{R}^3$, and pick two distinct points $a,\ b \in S$. Let $c$ be the set of points on $S$ equidistant to $a$ and $b$, where distance is defined by ...
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### Minimum distance between two arbitrary circles in space?

What is the minimum distance between two arbitrary circles in space? I am working out the problem with Maxima, but I am surprised by how complicated this rapidly turns out to unfold for such a "...
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### 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 ...