Questions tagged [computational-geometry]
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.
503 questions
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Cutting convex polygons into triangles of same diameter
This question continues from: Cutting convex regions into equal diameter and equal least width pieces
Definitions: The diameter of a convex region is the greatest distance between any pair of points ...
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finding a good term order for grobner basis
What are the tricks to pick a "good" monomial order to find a Grobner basis for a given ideal?
By good I mean one in which the final Grobner basis has a simple expression in terms of the ...
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Signed triangulations of simplicial polyhedra
Let $\partial S$ be the boundary of a compact polyhedron $S\subset\mathbb{R}^3$, assumed to be generic, in the sense that every face of $S$ is a triangle, and so that there are exactly two triangles ...
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Lower bound on the intersection of $\ell_1$ $n$-balls
Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ in $\ell_1$ norm, with distance $d$ and radius $R$.
Is there a lower bound on the volume of the intersection between the two n-balls? (assuming the ...
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Biggest Cartesian Product Included in a Real Plane Curve
Suppose an irreducible smooth $p \in \mathbb{R}[x_1,x_2]$ is given, and we would like to find finite sets $S_1 , S_2 \subset \mathbb{R}$ such that $p(S_1 \times S_2)=0$ and $|S_1 \times S_2|$ is as ...
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Optimally placing rectangles with obstacles
I am struggling with a fairly simple and natural geometric optimization problem, but I have not been able to find an obvious canonical method for solving it:
I am given a collection of $m$ axis-...
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Exact Value of a Constant Related to the Quickhull Algorithm
What is the exact value of the infinite sum
$$ \sum_{n=1}^{\infty}n2^n\sin\left(\frac{\pi}{2^n}\right)\left(1-\cos\left(\frac{\pi}{2^n}\right)\right)$$
That constant is related to the Quickhull ...
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Uniqueness of Riemann Constant Vector Solution
Let $X$ be a compact, genus $g$ Riemann surface (given as the desingularization and compactification of a plane algebraic curve), $J(X)$ its Jacobian, and $A : X \to J(X)$ the Abel map
$$A(P) = \left(...
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Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)
Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of $...
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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 ...
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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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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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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 ...
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Finding points inside innermost convex hull [closed]
Given a set of points $S$ on the Euclidean plane, Onion Peeling determines the nested set $H$ of convex hulls on $S$. Define an analytical formula on $S$ which produces a point, not necessarily in $S$,...
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find the collision of a particle with a swept triangle.
Given there is triangle: V in 3D space that transforms over time t -> t1 to V1, and a static point P is somewhere in 3d space, how can I determine if P ever collides with V, and if so at what value of ...
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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?
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Computer algebra for calculating curvature when the tensor metric is very big
Is there a computer algebra method to compute the curvature of a Riemannian metric on the plane when the metric tensor has long entries $E,F,G$
The computation by hand is very ...
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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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Show that the Minkowski sum of two triangles in 3D is the union of Minkowski sums of each triangle along the other's edges?
I'd like to show (or disprove) the claim that the Minkowski sum of two triangles with vertices in $\mathbb{R}^3$, $A+B$, is equal to the union of the unions of the Minkowski sums of $A$ along all ...
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Minimal degree of a polynomial such that $|p(z_1)| > |p(z_2)|, |p(z_3)|, ..., |p(z_n)|$
I was investigating the behavior of $p(x)^n \mod {q(x)}$, for some polynomials $p, q \in \mathbb{C}[x]$. We'll assume $q$ is squarefree. If $q(x) = (x - z_1) (x - z_2) (x - z_3) ... (x - z_n)$ for ...
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Discrete approximation of Minkshisundaram-Pleijel zeta function?
I'm looking for some references on the following situation:
$S$ is a Riemannian surface, and $G_n$ is a sequence of metric subgraphs embedded on $S$. Let $\zeta_n$ be the zeta function of the ...
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Covering the annulus of d-cube
Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c+x \mid c\in C \}$, for some $x\in R^d$, is called a homothetic copy of $C$. The ...
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Irreducibility of an explicit complex projective variety
Let $Y\subset \mathbb P^n_\mathbb C$ be a subvariety defined by a series of homogeneous polynomials $f_1, \ldots, f_t$. Is there an effective way to determine the irreducibility of $Y$ as an algebraic ...
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Convex polyhedra that can be folded from convex polygons
This question is based on http://www.science.smith.edu/~jorourke/Papers/FoldingPP.pdf.
Therein is stated the theorem: Every convex polygon folds to an infinite number (a continuum) of noncongruent ...
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Triangulations of point sets — obtuse and acute triangles
Given a planar configuration of points in general position. It is known that the Delaunay triangulation is the 'fattest' triangulation possible. It is also easily seen that given 7 points with 6 of ...
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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 ...
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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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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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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 plane ...
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Projection of a point to a convex hull in d dimensions
Hi,
I've got n points in d dimensions (typically n is around 30k-60k and d is 5 or 6). I'm using qhull to calculate the Delaunay triangulation and the convex hull of the set of points.
You can ...
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When are nontopological bistellar flips manifold-preserving?
A topological bistellar flip is the term used by Dougherty, Faber, and Murphy to describe a bistellar flip that does not cause any face of a complex to be duplicated. Suppose we consider a ...
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Planar convex region maximizing the difference in 'orientation' between its smallest containing rectangle and largest contained rectangle
We say a rectangle has orientation $\theta$ if the vector from its center to the middle of its shortest side (parallel to the longest side) has some angle $\theta$ with X axis.
Consider a planar ...
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Dispersion of a "random" subset of $[-1,1]^2$
Let $C$ be the square $[-1,1]^2$. Let $a_1,\dots,a_m$ be points chosen independently and uniformly at random from $C$. Let $d_m$ (dispersion) be the random variable $\max_{x \in C}{\min_{j \in [m]}{\|...
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Example of worst case distributions for 4D convex hull
My understanding is that convex hull of n points in 4D could have O(n²) edges in the worst case. Source: https://sites.cs.ucsb.edu/~suri/cs235/ConvexHull.pdf
This same source writes
In 4D, there are ...
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Do there exist smaller simplicial models of barycentric subdivisions?
Let $S$ be a simplicial complex and let $Bary(S)$ denote its barycentric subdivision.
Of course, the geometric realizations of $S$ and $Bary(S)$ are homeomorphic.
However, one issue that arises in ...
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'Constrained morphing' of planar convex regions
Morphing may be defined as a continuous transition of one shape to another. This post is about modifying planar regions continuously from one form to another under some constraints.
Qn: If $C_1$ and $...
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Determining if a polygon is convex using relations on orientation of each ordered triple of points
I am reading a paper by Szekeres and Peters on computing the 17-point case of the Erdős–Szekeres conjecture. The conjecture states that the minimum number of points in the plane (in general position, ...
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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 ...
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Choosing the weights of a Voronoi diagram -- is this function always the gradient of another function?
This question is related to the earlier question Weighted area of a Voronoi cell . As in that question, let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,...
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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 ...
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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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Sequence of polygons containing the shortest path
Hello all,
I’m looking at the weighted region problem i.e. trying to find the shortest weighted path across a polygon subdivision, but at this point in my work, I only need to know the sequence of ...
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Optimal unions of planar convex regions
This post continues Optimal intersections between planar convex regions.
Question: Given two planar convex polygonal regions $C_1$ and $C_2$, how does one algorithmically find how to place and orient ...
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Minimal vertices 3D polygon fitting between inner and outer boundaries
I have a set of 3D points representing a convex hull which I define as the inner boundary. The points are then offset outwards by a selected distance (the 'error') and I define this expanded hull as ...
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Existence of fine approximate of a convex body in $\mathbb R^d$ with convex hull of $\mathcal O(d)$ points
Let $K$ be a convex body in $\mathbb R^d$ which contains the origin and let $\theta \in (0,1)$.
Question. Is it always possible to find $n$ points $x_1,\dotsc,x_n \in \mathbb R^d$ such that
$$
\theta ...
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Existence of solutions of polynomials systems (and their "rough" shape) over $\mathbb{R}$ & friends with positive-dimensional ideals
This is a follow-up (but self-contained) question to my previous one. There I asked about state-of-the-art methods to solve multivariate polynomials systems over non-algebraically closed fields in ...
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Given a set of $n$ points in $[0,1)^d$, how do I partition the space into hyperrectangles such that each hyperrectangle contains exactly one point?
I'm new to this forum so I apologize if my question is ill-posed or too general. I have the following problem. Given a set of $n$ points in the unit hypercube, $[0,1)^d$, how can I partition the unit ...
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Build reversed No-Fit-Polygon
I need some robust algorithm to optimally fit one non-convex polygon into another. The destination one can contain holes.
Recently I found scholarly articles on this subject:
One of them describes ...
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2
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124
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Monotonicity for the side lengths of stars inscribed in regular polygons
Fix integers $l\ge 1$ and $n \ge 3$, and let $P_n$ denote the boundary of the regular $n$-sided polygon in the plane. We define a $(2l+1)$-pointed equilateral star to be a cyclically ordered list of ...
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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 ...
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