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3
votes
3answers
567 views

What fraction of a sphere's volume lies within a cone?

Let $B \subset \mathbb{R}^n$ be a collection of $n$ (not necessarily independent) unit vectors which we will label $v_1,\ldots, v_n$ for convenience. The cone $K_B \subset \mathbb{R}^n$ associated to ...
3
votes
1answer
267 views

Is list of distances defining points uniquely ?

There is N points on a plane. Is is feasible to reproduce there relative location having only list of distances. Assuming that translation, rotation and mirror are allowed in the result. The list ...
0
votes
1answer
330 views

In Taxicab Geometry, what is the solution to d(P, A) = 2 d(P, B) for two points, A and B? [closed]

Taxicab and Euclidean geometry differ a great deal, due to the modified metric function: dT(A, B) = |xa - xb| + |ya - yb| (Note that this means when measuring distance, it is not the length of the ...
8
votes
2answers
730 views

Altitudes of a triangle

The three altitudes of a triangle are concurrent -- this is true in all three constant curvature geometries (Euclidean, hyperbolic, spherical), but, as far as I know, the proofs are different in the ...
6
votes
3answers
976 views

Are properties of geodesics on a cylinder unique to cylinders?

The geodesics on a cylinder (a cylinder infinite in both directions) are either (1) simple (non-self-intersecting) closed geodesics, or (2) simple infinitely long geodesics (infinite in both ...
6
votes
2answers
181 views

Number of neigbour Voronoi cells for a random set of points on S^k or cube [-1, 1]^k?

Consider $S^k \subset R^{k+1} $. Sample $N$ points by say uniform distribution. (Example k=120, N=2^24, i.e. N>>k ). Consider Voronoi cell around each point. How many neighbours would a cell have ...
2
votes
2answers
488 views

Midpoint lattice polygons

Midpoint polygons (a.k.a Kasner polygons) have been studied, and their behavior is well understood. I am considering a variant, which I call midpoint lattice polygons. Start with a sequence of ...
16
votes
1answer
549 views

Is the ball reducible in some high dimension?

Let $K$ be a bounded symmetric ($-K=K$) open convex body in $\mathbb R^n$. The critical determinant $d(K)$ of $K$ is the least possible volume $|\operatorname{det}(a_1\dots a_n)|$ of the fundamental ...
11
votes
2answers
597 views

For what metrics are circles solutions of the isoperimetric problem?

A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this ...
1
vote
0answers
279 views

Simple development of simple curve on a cone

Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting) curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve $\overline{C}$ on a plane by rolling ...
2
votes
1answer
115 views

Graph implemened into the plane with segments as edges and we search for matching with no edges intersecting

There are some points in the plane and some of them are connected with segments between them. We look at this structure as a graph implemented into the plane where the points are the vertices and the ...
0
votes
0answers
158 views

Segmented area between circles

The following is a geometry problem that I came across with in the course of a research project. Consider a ray starting at some initial point $t$. Place point $s_1$ at distance $r$ from $t$ on the ...
0
votes
5answers
470 views

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 ...
2
votes
1answer
264 views

On distances between points on the plane

Take a set of 2n points on the plane and assume that no open set of diameter 1 contains more than n of these points. Question: con we pair up the points so that the distance between the points in a ...
0
votes
1answer
167 views

relation with jacobifields in a small neighbourhood

hi, I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
0
votes
1answer
103 views

4-polytope with vertices at the binary octahedral group

Hey everybody, Does anybody know if there is a convex polytope in $R^4$ with vertices at the binary octahedral group (identitfying $H$ with $R^4$). The binary tetrahedral group lies at the ...
5
votes
1answer
255 views

Geometric applications of Ekeland's variational principle

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself: Definition. Let $(X,d)$ be a ...
3
votes
0answers
63 views

Node-Weighted Euclidean Steiner Trees

I would like to know whether the following problem, including algorithms to solve it (exact or approximations) has been studied. A finite set of positive-weighted points are given in the ...
4
votes
1answer
223 views

Is it possible to sample uniformly on the surface of a high-dimensional polytope?

There are some pretty simple methods to do uniform sampling on the surface of high-dimensional spheres or cubes. Are there any methods that sample uniformly on the surface of a high-dimensional ...
16
votes
3answers
1k views

Prescribing areas of parallelograms (or 2x2 principal minors)

Let $(a_{ij})$ be a $n\times n$ symmetric matrix such that $a_{ij}\geq 0$ for all $i,j$ and $a_{ii}=0$ for all $i$. Under which conditions on the $a_{ij}$'s can one find $n$ vectors ...
5
votes
2answers
567 views

Pursuit-Evasion on a Manifold

I know pursuit-evasion has been studied in many contexts, including on a manifold (e.g., Melikyan, "Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds"), but I have not seen this version: ...
9
votes
1answer
1k views

What nets fold to polyhedra?

There is a classic (and open) problem asking whether every polyhedron can be unfolded to give a non-overlapping net. The converse problem has been studied asking which polygons can be folded in some ...
14
votes
5answers
818 views

How far is a set of vectors from being orthogonal?

Given some vectors, how many dimensions do you need to add (to their span) before you can find some mutually orthogonal vectors that project down to the original ones? Or, more formally... Suppose ...
9
votes
3answers
407 views

Recognising group actions on trees from the boundary

Let $G$ be a group acting on a locally finite tree $T$. Then the boundary $\partial T$ is a Cantor set on which $G$ acts by homeomorphisms (indeed by quasi-isometries under a suitable metric). ...
4
votes
2answers
467 views

Real vs complex surfaces

Hi! My background is in (complex) algebraic geometry. For the first time I'm considering varieties defined over $\mathbb{R}$ and I have some very basic questions about these. In particular I'm trying ...
5
votes
2answers
195 views

Normal form(s) for the elements of hyperbolic triangle groups

I'm seeking a reference or a sketch for any sort of normal form that would enable rapid enumeration without redundancies of the elements of hyperbolic triangle groups and/or von Dyck groups.
4
votes
1answer
2k views

Determine if you can build a polygon from segments [closed]

Is there a way to determine whether it is possible to build a polygon from given n segments? Maybe triangle inequality generalized?
4
votes
1answer
303 views

Polyline Averaging

I'm trying to find a method that can take a collection of polylines, each described by a list of connected points on a plane, and find an "average" path through them. The input lines do not loop. ...
5
votes
0answers
281 views

Maximum volume convex body coverable by a unit square

Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected) pieces (where $k=1$ means just the square). Your task is to construct the largest volume convex body ...
13
votes
3answers
2k views

What is the difference between holonomy and monodromy?

And what is the simplest example in which one is trivial and the other is not?
8
votes
1answer
1k views

The gimbal lock shows up in my quaternions

I suspect this is a bit basic for mathoverflow, seeing I'm still just an undergraduate I've been playing around with quaternions as means to eliminate the gimbal lock. From what I understand, one ...
11
votes
3answers
892 views

Covering a Cube with a Square

Suppose you are given a single unit square, and you would like to completely cover the surface of a cube by cutting up the square and pasting it onto the cube's surface. Q1. What is the largest ...
5
votes
2answers
355 views

Tangled Knot Function

I am seeking a function $f: \mathbb{R}^3 \to \mathbb{R}^3$ that has these properties: (1) When iterated $n$ times starting from some $p$, connecting the points in order with segments and closing ...
2
votes
2answers
290 views

Questions on calculating volume using n-1 forms

Is there an n-1 form on $R^n$ which calculates the volume of n-manifolds? Similarly, is there such a 1 form on $S^2$, and $RP^2$? I thought this has something to do with the orientation, is that ...
11
votes
2answers
468 views

What is the largest possible thirteenth kissing sphere?

It is well-known that it is impossible to arrange 13 spheres of unit radius all tangent to another unit sphere without their interiors intersecting. This was apparently the subject of disagreement ...
5
votes
3answers
361 views

Tetrahedron angles sum to $\pi$: Bisector plane

I discovered empirically what to me is an amazing lemma concerning face angles of a tetrahedron. Let $\triangle abc$ be a triangle in the $xy$-plane, and $d$ the apex of a tetrahedron with positive ...
0
votes
1answer
128 views

Space fill optimization

Are there pre-existing algorithms that will determine maximum number of possible polygons (rectangles only, or triangle only) that can "fill" a non regular area in 2D.
4
votes
3answers
1k views

minimum distance between two arbitrary circles in space

as per the title, i am working out the problem with maxima, but i am surprised by how complicated this rapidly turns out to unfold for such a "simple" question. monstrous equations, maybe someone has ...
10
votes
3answers
664 views

Optimal Wireframe Sphere

Suppose you have a length $L$ of metal pipe at your disposal, and you would like to build a wireframe unit-radius sphere, by bending, cutting, and welding the pipe into a connected structure $F$. Your ...
4
votes
1answer
208 views

Checking if one polytope is contained in another

Hi, I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other. At the moment I am ...
1
vote
2answers
1k views

Find the bounding box of a circle segment.

You have three points. A,B and C. They define a circle segment that starts at A, goes through B and ends at C. Find the smallest bounding box that encompases the circle segment. Here is a picture: ...
3
votes
2answers
531 views

Maximum area of intersection between annulus and circle? [closed]

Given two concentric circles $[C_1,C_2 ]$ with radii $(R_1 < R_2) $ creating an annulus; where should a third circle ($C_3$, radius $R_3$) be located such that the area of intersection between ...
4
votes
2answers
456 views

Area ratio of a minimum bounding rectangle of a convex polygon

Take a convex polygon $P$ in the plane, and find its minimum area bounding rectangle, $R$. I'm interested in the ratio of the area of $R$ to the area of $P$. The ratio has a minimum of $1$ for ...
7
votes
1answer
293 views

Generalizing a square wheel to a body rolling on a surface

A square wheel rolling on a catenary road maintains the wheel center at a fixed height, a well-known construction previously discussed on MO (e.g., "Generalizing square wheels rolling on inverted ...
7
votes
4answers
734 views

Geodesic path on the unit sphere with the sup norm

Let $X$ be the unit sphere of $\mathbb{R}^n$ with the sup norm, i.e. $X=\{x\in\mathbb{R}^n: \|x\|_{\infty}=1\}$. Let the metric $d$ on $X$ be the geodesic metric induced by the sup norm, i.e for any ...
2
votes
1answer
400 views

Filling $\mathbb{R}^3$ with skew lines

I would like to know if it is possible to fill $\mathbb{R}^3$ with lines with the following two properties: (1) Every point $x \in \mathbb{R}^3$ is contained in precisely one line. (2) Every ...
1
vote
0answers
335 views

Uniform convergence of convex functions - references

Inspired by the following question on stackexchange: http://math.stackexchange.com/questions/126142/uniform-convergence-of-convex-sequence-of-functions, I thought of asking whether anyone knows of ...
3
votes
1answer
349 views

In the classical construction of conic sections, where does the axis of the cone intersect the plane?

Everybody knows that if I take the intersection of a right circular cone with a plane, I get a conic section. My question is, where does the symmetry axis of the cone intersect the plane? Does this ...
1
vote
1answer
696 views

Covering an arbitrary polygon with minimum number of squares

I have a problem whereby, given an arbitrary polyogn with any number of points, I need to cover the whole area by a number of fixed size squares. I can easily find a set of squares which covers the ...
11
votes
5answers
2k views

Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...