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.
497
questions
3
votes
1
answer
99
views
Why is the Vietoris–Rips complex $\operatorname{VR}(S, \epsilon)$ a subset of the Čech complex $\operatorname{Čech}(S, \epsilon\sqrt{2})$?
$\DeclareMathOperator\Cech{Čech}\DeclareMathOperator\VR{VR}$I am reading Fasy, Lecci, Rinaldo, Wasserman, Balakrishnan, and Singh - Confidence sets for persistence diagrams (see here for a version of ...
8
votes
0
answers
175
views
Placing triangles around a central triangle: Optimal Strategy?
This question has gone for a while without an answer on MSE (despite a bounty that came and went) so I am now cross-posting it here, on MO, in the hope that someone may have an idea about how to ...
- 7,771
3
votes
1
answer
404
views
Detecting a PL sphere and decompositions
Question 1. A PL $n$-sphere is a pure simplicial complex that is a triangulation of the $n$-sphere. I'm looking for a computer algorithm that gives a Yes/No answer to whether a given simplicial ...
- 883
1
vote
1
answer
93
views
Algorithm to find largest planar section of a convex polyhedral solid
We add a bit more on shadows and planar sections following On a pair of solids with both corresponding maximal planar sections and shadows having equal area . We consider only polyhedrons.
Given a ...
- 5,827
1
vote
1
answer
128
views
An algorithm to arrange max number of copies of a polygon around and touching another polygon
A related post: To place copies of a planar convex region such that number of 'contacts' among them is maximized
Basic question: Given two convex polygonal regions P and Q, to arrange the max ...
- 5,827
1
vote
0
answers
114
views
Explicit computation of Čech-cohomology of coherent sheaves on $\mathbb{P}^n_A$
$\newcommand{\proj}[1]{\operatorname{proj}(#1)}
\newcommand{\PSP}{\mathbb{P}}$These days I noticed the following result of (constructive) commutative algebra, which I think is probably well known ...
- 650
5
votes
0
answers
413
views
Closest vertices of an AABB to a ray in n-dimensions
I came across this computational geometry problem and have not been able to find a satisfactory solution for it. A ray is known to originate from within an n-dimensional hypercube (AABB) in any ...
- 71
1
vote
0
answers
111
views
Inside-out dissections of solids -2
We record some general questions based on
Inside-out dissections of solids
Inside-out dissections of a cube
Can every convex polyhedral solid be inside-out dissected to a congruent polyhedral solid?...
- 5,827
3
votes
0
answers
68
views
Practical way of computing bitangent lines of a quartic (using computers)
Are there known practical algorithms or methods to calculate the bitangent lines of a quartic defined by $f(u,v,t)=0$ in terms of the 15 coefficients? Theoretically you can set up $f(u,v,-au-bv)=(k_0u^...
- 101
1
vote
0
answers
84
views
Inside-out dissections of a cube
Ref:
Inside-out polygonal dissections
Inside-out dissections of solids
Definitions: A polygon P has an inside-out dissection into another polygon P' if P′ is congruent to P, and the perimeter of P ...
- 5,827
5
votes
1
answer
251
views
Is the maximal packing density of identical circles in a circle always an algebraic number?
There is a lot of interest in the maximal density of equal circle packing in a circle. And I thought that knowing whether or not the solution is always algebraic or not would be useful.
My original ...
- 63
1
vote
1
answer
99
views
Finding the point within a convex n-gon that minimizes the largest angle subtended there by an edge of the n-gon
This post records a variant to the question asked in this post: Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon
Given a convex n-gon, ...
- 5,827
6
votes
2
answers
179
views
Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon
For any point P in the interior of a convex polygon, the sum of the angles subtended by the edges of the polygon is obviously 2π.
Given a convex polygon, how does one algorithmically find the point (...
- 5,827
0
votes
0
answers
29
views
Set of enclosed convex polyhedra in a graph
Given a straight-line graph embedded in $\mathbb{R}^3$ with known vertex coordinates and edges and no edge intersections, is it possible to find all the enclosed convex polyhedra inside? If so, is ...
- 1
1
vote
0
answers
91
views
An algorithm to decide whether a convex polygon can be cut into 2 mutually congruent pieces
This post is based on the answer to this question: A claim on partitioning a convex planar region into congruent pieces
A perfect congruent partition of a planar region is a partition of it with no ...
- 5,827
1
vote
0
answers
64
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 ...
- 111
0
votes
0
answers
64
views
Computational tasks resulting from Chern-Weil theory
I have recently learned Chern-Weil theory for smooth and complex manifolds, as well as surrounding material on cohomology with integral coefficients.
I am curious what computational tasks are ...
user30211
1
vote
1
answer
61
views
To maximize the volume of the polyhedron resulting from perimeter-halvings of a convex polygonal region
We add one more bit to Forming paper bags that can 'trap' 3D regions of max surface area (note: some possibly open related questions are also in the comments following the answer to above ...
- 5,827
1
vote
0
answers
90
views
A claim on the largest area circular segment that can be drawn inside a planar convex region
This post adds a little to To find the longest circular arc that can lie inside a given convex polygon
A circular segment is formed by a chord of a circle and the line segment connecting its endpoints....
- 5,827
0
votes
0
answers
62
views
Comparing partitions of a given planar convex region into pieces with equal perimeter and pieces of equal width
We continue from Cutting convex regions into equal diameter and equal least width pieces. There we had asked for algorithms to partition a planar convex polygon into (1) $n$ convex pieces of equal ...
- 5,827
1
vote
0
answers
81
views
Dissection of polygons into triangles with least number of intermediate pieces
This wiki article: https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien_theorem shows the dissection of a square into a triangle via 4 intermediate pieces. It appears easy to form a ...
- 5,827
2
votes
0
answers
65
views
Computational complexity of exact computation of the doubling dimension
Given a finite metric space $X$, the doubling constant of $X$ is the smallest integer $k$ such that any ball of arbitrary radius $r$ can be covered by at most $k$ balls of radius $r/2$. The doubling ...
1
vote
0
answers
41
views
Are there rectangles that can be cut into non-right triangles that are pair-wise similar and pair-wise non-congruent?
We generalize the questions of Can a square be cut into non-right triangles that are mutually similar and pair-wise noncongruent?
Can any rectangle be cut into some finite number of triangles that ...
- 5,827
1
vote
0
answers
92
views
Can a square be cut into non-right triangles that are mutually similar and pair-wise noncongruent?
We add a bit to Tiling the plane with pair-wise non-congruent and mutually similar triangles and Cutting polygons into mutually similar and non-congruent pieces
A (non square) rectangle can obviously ...
- 5,827
0
votes
1
answer
61
views
Robustness of doubling dimension to small perturbations
Let $M$ be a metric space. Then the doubling dimension of $M$, denoted $\dim M$, is defined to be the minimum value $k$ such that every ball in $M$ of radius $r$ can be covered by at most $2^k$ balls ...
2
votes
0
answers
89
views
To find the longest circular arc that can lie inside a given convex polygon
Question: Given a convex polygonal region P, to find the longest connected subset of a circle that can lie entirely in P.
For some P, the optimal subset will be a full circle; otherwise, a single arc ...
- 5,827
2
votes
1
answer
259
views
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 ...
- 279
2
votes
0
answers
65
views
Is this an actual solution for centroidal Voronoi tiling, or just a visual approximation? [closed]
For the capstone project at the end of my graduate Data Science studies in late 2022, I needed an algorithm that would converge to something close to centroidal Voronoi tiling, while tiling contiguous ...
- 121
0
votes
0
answers
23
views
Smallest Semi-ellipses that contain a convex n-gon
Definition: Let us define a semi-ellipse as either of the two halves into which an elliptical region is cut by any line that passes through its center. Obviously, all semi-ellipses that come from any ...
- 5,827
0
votes
0
answers
50
views
Optimal packing and covering of a triangle with squares
We continue from Another variant of the Malfatti problem.
Given a triangle T and a number n, how to cover it with n squares (of possibly different dimensions) such that the sum of the areas (...
- 5,827
2
votes
0
answers
127
views
Graph Laplacians, Riemannian manifolds, and object collisions
To preface this question, I am a part-time game developer and full-time optimization fiend.
I am working on object collisions at the moment and many resources I have found online are more-or-less just ...
- 21
4
votes
2
answers
201
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 ...
- 142
0
votes
0
answers
63
views
Bounds for the Dispersal Problem in convex regions
We add a bit to: Bounds for minimax facility location in a convex region
Two earlier posts: Cutting convex regions into equal diameter and equal least width pieces - 2 and Facility location on ...
- 5,827
1
vote
0
answers
89
views
Bounds for minimax facility location in a convex region
An earlier question: Facility location on manifolds
A possibly related earlier post: Cutting convex regions into equal diameter and equal least width pieces - 2
The minimax facility location problem ...
- 5,827
3
votes
0
answers
178
views
Algorithm to dissect a polygon into a minimum amount of rectangles, conditioned on a maximum overlap
I have the following problem, I have a problem regarding concave polygons. I want to write code to cover any polygon with a minimum amount of rectangles that are allowed to overlap and have no fixed ...
- 31
2
votes
1
answer
190
views
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 ...
- 3,049
2
votes
0
answers
103
views
Description of a point cloud being "undersampled" wrt persistent homology, confidence level?
I am completely new to topological data analysis, so I apologise if this is a well-known area of persistent homology, as well as for any imprecise language.
Suppose we know completely the topological ...
- 21
3
votes
1
answer
133
views
Computer program for polyhedral manifolds
Suppose I have a 3-manifold obtained via face identifications of a polyhedron (e.g. the Poincaré sphere presented as a dodecahedron with opposite faces glued). Is there a program that exists for ...
- 155
0
votes
0
answers
45
views
Persistent diagrams for images : existing implementations or packages?
I am interest to compute the persistent diagram associated to the image of a persistent module as in ''Persistent Homology for Kernels, Images, and Cokernels'' : https://epubs.siam.org/doi/epdf/10....
- 73
1
vote
1
answer
56
views
On largest convex m-gons contained in a given convex n-gon where m < n
This post is the inside-out variant of On smallest convex m-gons that contain a given n-gon where m<n
Given a convex n-gon region P, and an m less than n, how to find the max area convex m-gon Q ...
- 5,827
0
votes
0
answers
91
views
On smallest convex m-gons that contain a given n-gon where m<n
Given a convex n-gon region P, and an m less than n, will the least area convex m-gon Q that contains P be such that an edge of Q coincides with an edge of P (in other words Q cannot be such that P ...
- 5,827
6
votes
1
answer
356
views
Desargues ten point configuration $D_{10}$ in LaTeX
I want to draw the Desargues configuration $10_3$ in LaTeX using the standard picture environment, which allows only lines with the slopes $n:m$ where $\max\{|n|,|m|\}\le 6$. Is it possible? If not, ...
- 41.1k
2
votes
0
answers
112
views
Understanding normalization algorithms
Let $R$ be a commutative and reduced ring, finitely presented over $\mathbb Z$. Let $\overline R$ be the integral closure of $R$ in its total ring of fractions. In https://arxiv.org/abs/alg-geom/...
- 211
2
votes
1
answer
130
views
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 ...
- 5,827
1
vote
1
answer
78
views
To optimally wrap convex laminae with paper
Ref: On folding a polygonal sheet, Multi-layered wrapping of polyhedra
Basic intent: to wrap a given convex planar lamina with a convex sheet of non-stretchable paper (such that every point on both ...
- 5,827
1
vote
0
answers
34
views
On partitioning convex polygonal regions in area ratio $t : (1-t)$ where $0<t<1/2$ with least length of cut
Question: Given a convex n-gon P. How can we efficiently find the partition of P into 2 pieces with areas in the some given ratio $t : (1-t)$ where $0<t<1/2$ such that the length of cut is ...
- 5,827
1
vote
2
answers
116
views
Are there variants of Euclidean Steiner Tree problem that are known to be in P?
Question: The Euclidean Steiner Tree problem (https://en.wikipedia.org/wiki/Steiner_tree_problem) is NP hard. Are there non-trivial (constrained) variants of this question that are known to have ...
- 5,827
1
vote
1
answer
72
views
Partitioning polygons into obtuse isosceles triangles
Ref:
Partitioning polygons into acute isosceles triangles
Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
https://math.stackexchange.com/questions/1052063/...
- 5,827
1
vote
0
answers
58
views
Covering a unit square with odd number of equal area triangles - optimally
We add a bit to this post: Cutting off odd numbers of equal area triangles from a unit square
Question: Given an odd integer n, how does one cover the unit square completely with n equal area ...
- 5,827
3
votes
2
answers
226
views
Writing a smooth plane quartic as the vanishing of $Q_0Q_2 - Q_1^2$ for quadratic $Q_0,Q_1,Q_2$
It seems well-known that any smooth plane quartic can be written as the vanishing of $Q_0Q_2 -Q_1^2$. Is there a good way to work out these quadratic factors $Q_0,Q_1,Q_2$? For example, given the ...
- 557