Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,403 questions
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When is the conical hull of a finite set of vectors a subset of the space? (and tilings)
Consider a hypercube in n-dimensions, and take some projection down to an m-dimensional subspace. Now take all vertices and m-1 dimensional facets visible from some direction outside the projection. ...
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Fitting an ellipse to an arbitrary polygon
Hello,
I'd like an algorithm for fitting an ellipse to a polygon. This polygon may be convex or concave. I've read about fitting an ellipse inside or outside a polygon (maximal and minimal of size, ...
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785
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A curious property of the Gergonne point
Ha, finally no knot theory :-)
First of all, let's define the "power line" of three circles.
(Very probably, someone had the idea before me, but no math forum
ever came up with something .)
Call the ...
- 4,829
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246
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How to derive an energy measure of metric deforming
The problem is an abstract from applied science.
Given an $n$ dimensional Riemann manifold with metric $\langle M, g\rangle$, we could define deformation of the metric $g(t)$ where $t\in [0,1]$, for ...
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Are spherical codes algebraic?
Jeffrey Wang in Section 4.2 writes "Since a code is the solution to a number of polynomial equalities between the shortest edges, the coordinates of each rigid point in the code are algebraic and lie ...
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Terminology: metric space with product and unit, and the opposite of a nonexpansive map
Someone I know is trying to figure out if the following concepts already have an established name in the literature, and MO is a great place to ask around.
1) Suppose $X$ is a metric space equipped ...
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335
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Irrationality of square root of 2 [closed]
It is possible to explain me the 18th proof of the irrationality of square root of 2 from the following site?
http://www.cut-the-knot.org/proofs/sq_root.shtml
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Cluster-preserving and distance-maximizing embedding into Hamming Space?
I have a set of data, each instance in the real $[0,1]^{d}$. However, it's actually all in a relatively small range around 0.5, clustered into classes in even smaller ranges. The actual origin of the ...
- 2,383
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3D Delaunay Triangulation -> Euclidean Minimum Spanning Tree
I read that the Euclidean Minimum Spanning Tree (EMST) of a set of points is a subgraph of any Delaunay triangulation. Apparently the easiest/fastest way to obtain the EMST is to find the Deluanay ...
- 171
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Characterization of angles trisectable with straightedge and compass
Lindemann's proof of the transcendence of $\pi$ has settled the question, whether an arbitrary angle can be trisected, using straightedge and compass alone, to the negative.
In the following, ...
- 13.2k
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Is the intersection of two minimal surfaces minimal?
Consider two $n$-dimensional minimal surfaces without boundary. Suppose they are embedded in some $\mathbb{R}^m$ in such a way that they intersect transversally. Is their intersection a minimal ...
- 1,703
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524
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Preservation of injectivity radius
Suppose I have a smooth, noncompact manifold $M$ with metrics $g_i$ for $i = 1,2$. Suppose there exists a $C \geq 1$ such that
$$ C^{-1} d_1(x,y) \leq d_2(x,y) \leq C d_1(x,y)$$
where $d_i$ are the ...
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457
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Confining a polytope to one side of an affine hyperplane
Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem.
This answer on math.stackexchange.com claims the ...
- 2,239
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129
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Planar curves identical to their inverses
Is the right strophoid
the only planar curve $C$ whose inverse curve w.r.t. some circle (in this case: centered on the origin)
is identical to $C$?
&...
- 151k
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5
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Generate points of a (n-2)-sphere on a n-hyperplane [duplicate]
Possible Duplicate:
Efficiently sampling points uniformly from the surface of an n-sphere
I'm trying to generate random points of a (n-2)-sphere on a n-hyperplane so basically the intersection of ...
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476
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coordinate free foundations of trigonometry [closed]
What axioms for geometry and trigonometry would I have to chose in order to completely avoid coordinates in defining trig functions and showing the equivalence of their geometric (unit circle) and ...
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346
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AC and Euclidean Geometry [closed]
It there any relation between the axiom of choice and Euclidean Geometry ??
I mean what are the known statements, theorems or results in euclidean geometry that are dependent on AC ?? (this question ...
user30669
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What is the smallest area of a central section of the unit hypercube?
Let $\mathcal{U} \subseteq \mathbb{R}^n$ denote the unit hypercube i.e. $\mathcal{U} = [0,1]^n$, and assume that for some $d \in \mathbb{R}^n$ one denotes by
$$
\mathcal{H} = \left\{x \in \mathbb{R}^n ...
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357
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Metric space that is not a subspace of $\mathbb{R}^n$? [closed]
What are some simple examples of metric spaces that cannot be subspaces of $\mathbb{R}^n$? I've heard there is an example with $4$ points, where two points lie between the other two, but I cannot ...
- 1,465
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Two questions about Convex Sets and Lebesgue Measure
For any positive integer n, let E(n) denote n-dimensional Euclidean Space and let L(n) denote n-dimensional Lebesgue Measure on E(n). Take n=2 for simplicity. There are uncountably many convex subsets ...
- 6,215
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Categories of Geometry [closed]
I learn that Geometry has several categories/subfields from Wikipedia. But I am still not clear about the standards according to which they are classified.
It seems Euclidean Geometry, Affine ...
- 357
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94
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Kähler metric on the projective space
"Is there a Kähler metric on the complex projective space $\mathbb {P} ^n(\mathbb {C} ) $ different from the Fubini-Study metric?
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175
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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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937
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The exact number of points within a circle of radius r centered on a lattice point in a hexagonal lattice? Review expression Gauss circle problem
In the case of a square lattice, the exact number of points within a circle of radius r centered in the center is (see: http://mathworld.wolfram.com/GausssCircleProblem.html:
$$N(r)=1+4Floor(r)+4 \...
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238
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Does anyone know of an academic reference for this proof that the tangent to a hyperbola at a point bisects the angle between each focus to the point?
I am writing a report and as part of it I need to prove the property that for a point $P$ on a hyperbola, the tangent to the hyperbola at $P$ bisects the angle $\angle F_1PF_2$ where $F_1$ and $F_2$ ...
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1
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228
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Uniform distance from a discontinuous function is continuous
Define the metric $d(f,g)\triangleq \sup_{x \in [0,1]} \|f(x)-g(x)\|$ on the set $\operatorname{B}$ of uniformly bounded functions from the interval $[0,1]$ to $\mathbb{R}$, fix $g \in \operatorname{B}...
- 5,405
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Convex planar curves and intersections [closed]
Given two planar regular convex not-closed curves C and C_1.
Let A the set of finite intersections between C and C-1.
Then what is the stricter upper bound of |A|?
I would say 2.
Thanks.
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Is this bounded?
May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a
convex polygon in the plane and $v_{m+1}$ be a vertex in the interior
of the convex polygon. Connect ...
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2
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6k
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Best fit circles inside a square based on side of the square [closed]
Hi,
How to calculate number of circles in side a square, if we know the side of the square and the circles all are equal size.
Thanks.
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What happens to the volume of an ellipsoid as the number of dimensions increases? [closed]
Would the volume of an ellipsoid continuously increase if one keeps adding radii along new dimensions? What is the volume of ellipsoid with infinite dimensions?
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How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature
Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
- 1,219
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115
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Generalized Triangle Inequality for Snowflakes
Let $p>0$ and consider a metric space $(X,d)$. I have recently come across a problem where the space $(X,d^q)$ provides is natural; where $q>1$. However, the triangle inquality break (i.e. it ...
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Does an affine building associated to a group satisfy the axioms of building?
Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\...
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514
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Distance between two points using triangulation
Suppose we have two points $p_1$ and $p_2$ in a metric space with unknown dimensionality, with no way to directly compute the distance between them, e.g. no coordinates.
Say we can randomly sample a ...
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1
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407
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Parametrization of polygons and polyhedra [closed]
So, I've got a pretty interesting problem:
I was wondering how one would go about trying to generate every n-gon, or at least parametrize the space of a specific n-gon (say a hexagon) so it's easily ...
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1
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1k
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Example distance metric that is not conditionally negative definite
Theorem 4.1 of this paper says that there exist distance matrices that are not conditionally negative definite (CND). How do I construct an example of a distance matrix that is not CND? Do you know an ...
- 399
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Determine the boundary points of a set of points [closed]
I have a set of points $S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$. Then how to find the boundary points (which is a subset of $S$) of $S$?
There are methods like convex hull, concave hull and $\...
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136
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What is the area of the piece of an $n$-sphere within a given angle of a vector? [closed]
Let $x$ be the unit vector $(1,0,0,\ldots,0)$ in $\mathbb{R}^n$, and let $A(\theta)$ be the subset of $\mathcal{S}^{n-1}$ whose angle to $x$ is less than $\theta$, i.e.
$$ A(\theta) = \left\{ y \in \...
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Quaternion between two quaternions [closed]
Hello,
I have an orientation P1 in a 3D space, represented as a quaternion [w x y z].
Then P1 is rotated using another quaternion (q1) with the formula
P2=q1*P1*q1'...
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253
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Equidistant points in negatively curved metric spaces
Suppose that $X$ is a simply connected metric space, with a non-positively curved metric (for example, Euclidean or hyperbolic space). Let $A,B,C$ be disjoint, convex sets in $X$, and suppose that the ...
- 1,329
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96
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Conditions on a parametric curve so that its normal plane covers R^n
I am working on a control theory problem that either just caught me on a blind spot or is beyond me. I guess it's not a new question, but I couldn't find anything about it.
Let $p(s), s\in \mathbb{R}, ...
0
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1
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69
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Is there a method to make a rep-n rep-tile for any number n, using only triangles?
Is there a method to make a rep-n rep-tile for any number n, using only triangles? And if there is no such method, what's the smallest number for which there's no example?
I'm only considering rep-...
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1
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277
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Projection of convex set onto a convex set [closed]
Can projection of convex sets onto convex sets be non-convex yet connected? If so is there any necessary and sufficient conditions?
Can projection of $n$ dimensional convex sets in $\mathbb R^{n'}$ ...
- 13.9k
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1
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117
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Neighborhood of $(n,\delta)$-strained point in Alexandrov space homeomorphic to $\mathbb{R}^n$, how big is $\delta$?
Let $M$ be an $n$-dimensional Alexandrov space with curvature $\geqslant k$. A point $p\in M$ is said to be an $(n,\delta)$ strained point if there are $n$ pairs of points $a_i, b_i$ such that
$$
\...
0
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1
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58
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How to compute the parameters of circumscribed hypershpere? [closed]
Assume I have an $n$-dimensional simplex on the points $x_0, ..., x_n$ where each $x_i \in \mathbb{R}^n$. I would like to obtain the parameters (center and radius) of it's circumscribed n-dimensional ...
- 283
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335
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What conditions are sufficient to prove a transformation in $\mathbb{R}^2$ is affine? [closed]
What conditions are sufficient to prove a transformation in $\mathbb{R}^2$ is affine?
I currently have shown that the transformation is bijective, points and lines are preserved and also the ...
- 109
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3
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554
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Converting a bounded metric into an unbounded metric
Suppose $d$ is a bounded metric on $X$, i.e. $d(x,y)< K<\infty$ for all $x,y\in X$. Is there a standard way to convert $d$ into another metric $\widetilde{d}$ on $X$ with the property that $\...
- 710
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128
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What does it mean exactly for a pair of $S^0$'s to be unlinked on a knot $K$?
I am trying to learn about the effects of knot mutation on the hyperbolic manifolds obtained via hyperbolic Dehn surgery, and I'm currently reading Ruberman's paper "Mutations and Volumes in $S^3$" (...
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1
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126
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When are the main diagonals of a convex $2n$-gon concurrent?
By main diagonals I mean the diagonals $A_iA_{n+i},$ of which there are $n.$ One classical result in the hexagonal case is that this is true for cyclic hexagons with $ace = bdf.$ I'm wondering when ...
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