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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Graph drawing: unrooted undirected tree graphs with specified edge lengths
Has Joseph Felsenstein's equal daylight layout been analyzed by the graph drawing community? The following description is taken from his
drawtree
documentation (Wayback Machine):
"This ...
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0
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Computational hardness of a discrete generalized rectangle packing problem
I have a decision problem that is clearly in NP, but I cannot seem to prove that it is in P, nor can I prove its NP-hardness. I attribute this more to my inexperience than to the problem's difficulty (...
3
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1
answer
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Pointers/Papers on subdivision of planar quadrilateral meshes (PQ-Mesh) in 3D?
I'm interested in the subdivision of planar quadrilateral meshes (PQ-Meshes). Meshes consisting only of planar quadrilaterals, like discrete Voss surfaces and alike. I've been searching the web
for ...
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2
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A quadratic $O(N)$ invariant equation for 4-index tensors
Consider an $O(N)$ invariant quadratic equation
$$
T_{ijkl}= T_{ijmn}T_{klmn}+ T_{ikmn}T_{jlmn}+ T_{ilmn}T_{jkmn},
$$
where $T_{ijkl}$ is a real, totally symmetric 4-tensor, and the indices run from 1 ...
1
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0
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Upper bound on the diameter of a convex lattice n-gon with a given area
Given the area $A$ of a strictly convex polygon with $n$ vertices with integer Cartesian coordinates, there are usually several non-equivalent polygons. The relationship between the area, the number ...
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Partition of polygons into 'strongly acute' and 'strongly obtuse' triangles
Definition: Let us refer to obtuse triangles with the largest angle strictly above a given cutoff value as 'strongly obtuse' - the definition is parametrized by the cutoff value. Likewise, strongly ...
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4
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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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Boolean operation on n dimensional polyhedron
A polyhedron in $R^n$ is defined by a set of half-planes: $P = \{x \in R^n \mid Ax - b \le 0\}$.
Given a set of polyhedra in $R^n$, $ P_1, P_2, \dotsc, P_k$, is there an algorithm/implementation that ...
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Are point sets of the same order type connected by continuous (order type)-preserving motion?
Given two general position point sets in $\mathbb{R}^2$ of the same size and order type, is it possible to continuously move the points of one set until they coincide with those of the other set in ...
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Translate of a geodesic that goes through a fixed point on $\mathbb{H}$
Consider the complex upper half plane $\mathbb{H}$ with the hyperbolic geometry. Fix a point $z \in \mathbb{H}$ and also a geodesic $c$. I want to find a hyperbolic translation $\gamma c$ passes that ...
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Ellipse of least perimeter that contains a given triangle
This post is related to Smallest 3-ellipses that contain triangles and tries to clarify a basic issue.
Question: Given a general triangle T, How does one find and characterize the ellipse of least ...
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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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answers
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Dissection of polygons that preserve boundaries
Ref:
Inside-out polygonal dissections
Further queries on inside-out polygonal dissections
Question: Consider any two polygons P1 and P2 with equal area and equal perimeter. Is it always possible to ...
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3
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$n$-dimensional Voronoi diagram
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 it;
b)...
- 63.9k
2
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1
answer
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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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Optimal intersections between planar convex regions
Here is an earlier discussion that could be related:
On comparing planar convex regions of equal perimeter and area
We are broadly interested in placing two given planar convex regions so that the ...
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1
answer
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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 ...
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Partitioning convex n-gons into least number of equal area convex quadrilaterals
This post adds a bit to Partitioning convex polygons into quadrilaterals of equal area and perimeter
Question: How does one achieve the partition of any given convex n-gon into the least number of ...
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answers
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The fastest way to sample points on an implicit manifold, or projecting points on a manifold
Given a compact manifold $M$ in $R^n$, $M = f(x)$, f(x) is infinitely differentiable. $x$ $\in$ $R^n$, I want to find a bunch of samples on the manifold.
Currently, I'm setting up an SQP optimization ...
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1
answer
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How to efficiently vacuum the house
Let $P$ be a polygon (perhaps with no acute angles inside) and let $L$ be a line segment. The segment may move through the area inside $P$ in straight lines, orthogonal to $L$, or it may pivot on any ...
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Complexity of counting regions in hyperplane arrangements
Let $H_1,\ldots,H_n$ be hyperplanes in $\Bbb R^d$. Denote $\mathcal{H} :=\{H_1,\ldots,H_n\}$ and let $c(\mathcal{H})$ be the number of regions in the complement: $\Bbb R^d\setminus \bigcup H_i$.
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What are some other methods for partitioning an n-dimensional space based on a set of points in that space?
So this is a very general question, but I'm curious if there are any other methods for partitioning an n-dimensional space based on the location of a set of points, either randomly chosen or specified,...
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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 ...
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2
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1
answer
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views
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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2
answers
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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 ...
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2
answers
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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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1
answer
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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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Reference request: How to construct a diffeomorphism between point clouds
I'm interested in the following question:
Given two sets $S = \{x_1, ..., x_N\}$ and $T = \{y_1, ..., y_N\}$ each consisting of $N$ distinct points in $\mathbb{R}^n$, how can we construct a ...
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1
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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 ...
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answer
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What can be said about the Shadow hull and the Sight hull?
This is a question implicitly raised by Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always ...
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0
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Cylinder orientation representation
I'm trying to find an efficient computation and representation for the following problem.
Given a cylinder with height $h$ and radius $r$ with a given position $\mathbf{x} = [x, y, z]$ and $N$ number ...
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Is a given point in the interior of the convex hull of a given finite collection of points?
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 ...
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0
answers
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Fermat point amidst polygonal obstacles
Consider $k$ distinct points in 2D-plane with $n$ convex polygonal obstacles. Is there a poly-time algorithm (poly in $k$ and the total number of obstacle vertices) to find a point outside of all ...
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Reduced Voronoi diagram
I am currently reading Differentiable Surface Triangulation presented at Siggraph Asia 2021.
I think most of the paper is clear to me, though I keep re-reading through to see if I miss details.
The ...
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0
answers
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Explicit computations of finite covers of genus one curves with two points of ramification
I have an explicit genus one curve $E$ with two points $p_1$ and $p_2$ on it and am looking for an explicit degree seven cover $X\to E$ with ramification precisely over $p_i$, with a single preimage ...
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Are cells of 4-polytopes a convex polyhedron by definition?
I'm going by the Wikipedia definition for a 4-polytope.
Do by definition, cells of 4-polytopes have to be a convex polyhedra?
If not, then are there polyhedra with non-convex faces?
If yes, is it the ...
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2
answers
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Covering a polygon with rectangles
I am trying 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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What is the VC-dimension of regular convex k-gons in the plane?
Recall the relevant definitions:
Let $H$ be a family of sets in $\mathbb{R}^d$. The intersection of $H$ with a point set $C$ is defined as $H\cap C = \{h\cap C\mid h\in H\}$. The VC-dimension of $H$ (...
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Angles between edges of a geometric graph and graph invariants
Are there any clever ways in which the angles between edges in a geometric graph are encoded in the graph spectrum, or another object associated with the graph?
I'm interested to see what else is ...
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How to do an elevated 2D Delaunay triangulation?
This is what I call an elevated Delaunay triangulation:
This is also called a 2.5D Delaunay triangulation.
To do it, I simply perform an ordinary 2D Delaunay triangulation with the (x,y)-coordinates, ...
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Comparing convex planar regions of equal perimeter and area - 2
We try to extend On comparing planar convex regions of equal perimeter and area .
Given two planar convex regions C1 and C2 both with unit perimeter, we define the difference between C1 and C2 as the ...
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A variation on the projective Nullstellensatz
Let $V$ be a $\mathbb{C}$-vector space, and let $f_1,\dots,f_n \in S^d(V^*)$ be homogeneous polynomials of degree $d$ for which $V(f_1,\dots, f_n)=\{0\}$.
Must there exist a positive integer $k\geq d$ ...
- 148k
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votes
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answer
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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
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votes
1
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Finding the smallest centrally symmetric region that contains a convex planar region
Given a convex polygonal region C, how does one find/characterize the smallest zonogon (centrally symmetric convex polygon https://en.wikipedia.org/wiki/Zonogon) that contains C?
Note 1: In question ...
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1
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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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2
answers
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Finding a point inside a surface
I have a triangulation of a surface without boundary in $\mathbb{R}^3$.
The triangulation gives a unit normal pointing outwards for each triangle. I need to find some point in the interior of the ...
CommunityBot
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A ratio to measure 'roundedness' of planar convex regions
Ref: A center of convex planar regions based on chords
The above discussion quotes the definition of 'centralness coefficient' and defines a center of a planar convex region. 1/2 is the least possible ...
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A center of convex planar regions based on chords
This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions.
A point $P$ in the interior of a planar convex region $C$ divides ...
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answer
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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 ...
- 10.6k
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5
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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 ...
