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2answers
3k 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
720 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
572 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
341 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 ...
8
votes
4answers
852 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
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1answer
512 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
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0answers
488 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
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1answer
403 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 ...
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1answer
1k 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 ...
13
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5answers
3k 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 ...
4
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1answer
331 views

Least cardinality of a set of points in the plane

What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all ...
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1answer
279 views

Manifold with no Finsler structure?

Is there a good example for a smooth manifold to which one cannot give a Finsler structure in any meaningful way? Ideal the example should be of low dimension and not too bizarre.
5
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1answer
743 views

theorems equivalent to the parallel postulate

Is there a good survey article listing all the theorems of Euclidean geometry that are equivalent to the parallel postulate?
2
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0answers
271 views

Minimum solid angle and aspect ratio of an $n$-simplex

In computational geometry and other fields, it is of interest to have degeneracy measures for shapes of simplices, which quantitatively seperate the regular simplex from degenerate simplices. In two ...
1
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2answers
330 views

Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron

I've been trying to find, without much success, 4 triangles whose corresponding sides are congruent that cannot be folded into a tetrahedron. Anyone has any clue how to approach this problem?
5
votes
2answers
440 views

Conformal structure does not see conical singularities

the conformal structure does not see the conical singularities of a polyhedral surface. This is a quote from the Preface of Quantum Triangulations (eds.: Carfora, Marzuoli). The sentiment is ...
1
vote
1answer
285 views

triangle equality in manifold

For a generilized triangle on a manifold, (distance can be regarded as geodesic length)it is well known that for Eucilidean Geometry,the following is true: Consider a triangle $ABC$, $D$ is the ...
1
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2answers
267 views

Smooth a matrix

I have a matrix in which each element contains the coordinates of a 3D surface. Sometimes, some points will be "out of line" meaning that they will not conform to the general shape. For example you ...
0
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1answer
415 views

Action of Isometries on a Line in the Plane

I'm trying to determine the stabilizer of a line in a plane when acted upon by the group of isometries of the plane. Please note that I'm using the notation found in the Wikipedia article on Euclidean ...
0
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2answers
246 views

Derivation of surfaces

In an Euclidean linear plane, the evolute of a given curve C with support function h(t) can be regarded as a kind of derivative C' of C. Indeed, C' has support function h'(Pi/2 -t). Is there any ...
3
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0answers
131 views

The mean number of vertices in small connected components of random geometric graphs

I place $N$ points on a circular plane of radius $R$, and draw edges to connect points that are less than or equal to some distance $D$ to form a set of graphs or cliques $G_i$. As a function of $N$, ...
1
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1answer
237 views

A raceway problem

Let $f(x)=\sin x$, and $g(x)=\sin x + 1$. Consider a set $S=\{(x,y)| f(x)\leq y \leq g(x), x\in [0,2\pi]\}$. This set $S$ can be considered as "Raceway" My question is finding the shortest path in ...
6
votes
2answers
881 views

Finding the convex combination of vertices which yields an inner point of a polytope

Given a convex polytope $P\in \mathbb{R}^n$, and a point $x\in P$, Caratheodory's theorem gives us that there exists a set of at most $n+1$ vertices of $P$, such that $x$ is a convex combination of ...
10
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1answer
574 views

Polygons uniquely inducing arrangements

A beautiful, relatively recent result is that, Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$. In a simple arrangement, every pair of lines ...
9
votes
2answers
594 views

Ellipsoids and lattices: an enclosure problem.

$E \subset {\mathbb R}^2$ is an ellipse of area $1$ centered at the origin that contains no other point with integer coordinates. Is there a matrix $A \in SL(2,{\mathbb Z})$ such that the ellipse ...
3
votes
4answers
1k views

space of geodesics

hallo, i have the following problem: Let $(M,g)$ be a compact Riemannian manifold with metric $g$ and $\nabla$ be the Levi-Civita Connection. Denote by $G(M) =${$\gamma: \mathbb{R} \rightarrow M | ...
2
votes
1answer
200 views

Locus of points where difference in gravitational forces is constant

Is there a name for the curve in the plane defined by $a/\|x - p\|^2 - b/\|x - q\|^2=\mathrm{constant}$ where $a$ and $b$ are fixed numbers and $p$ and $q$ are fixed points? How about if I don't ...
9
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4answers
431 views

Can one do without a classifying space when showing vanishing of cohomology

Let $G$ be a discrete group and $A$ an abelian group, then $H^n (G,A)$ can be defined as $$ H^n (G,A) = H^n (B_G, A)$$ Where $B_G$ is the classifying space of $G$, i.e. $B_G = E_G / G$ where $E_G$ is ...
3
votes
2answers
817 views

absolute continuity on $R^{n}$

I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$. I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
0
votes
1answer
2k views

Tangent lines to 2 circles, tangent planes to 3 spheres, and so on.

Although it is known the solution to the first two questions, somebody may have different nice answers, so I include them: Given two circles in the plane, there is (at least) a line which is tangent ...
14
votes
1answer
573 views

Ratio of circumscribed/inscribed $(n{-}1)$-gons

As a discrete analog of the MO question, "Löwner-John Ellipsoid: incribed and circumscribed," I've been wondering what might be the maximum ratio of this quantity? Let $P$ be a convex polygon of $n$ ...
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votes
2answers
585 views

projective camera: back-projecting a point on the image plane into 3-space

suppose I got a projective camera model. for this model I would like to back-project a ray through a point in the image plane. I know that the equation for this is the following: $$ y(\lambda) = P^+_0 ...
5
votes
0answers
284 views

Local minimum from directional derivatives in the space of convex bodies

I have a function $f(K)$ defined on the space of three-dimensional convex bodies for which I want to show that the unit ball $B$ is a local minimum. I have been able to show if $K$ is not homothetic ...
5
votes
1answer
501 views

Bending an elastic, inextensible sheet of paper into a teardrop shape

I take an rectangular sheet of paper, of height $H$ and Young's Modulus $E$, and in the absence of gravity, I bend it into a "teardrop" shape so that the edges along the top and bottom touch only ...
4
votes
0answers
93 views

Applications of k-medians with moment constraints

Suppose I have a collection of points $p_1,\dots,p_N$ in the plane. In the Euclidean $k$-medians (or $k$-means) problems, our objective is to distribute a set of $k$ points $x_1,\dots,x_k$ in the ...
10
votes
2answers
479 views

The “grassmannian” of a simplicial complex

This question is mainly a reference request – I have a construction which seems natural, so I am quite convinced it should be standard, but I don't know what it is called. Take an $n$ dimensional ...
1
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0answers
114 views

Proving that an optimal solution “converges”

This question is a follow-up on a previous question I asked at: Distances between and among points in a region Let $X = x_1,\dots,x_n$ denote a finite set of $n$ points in the unit circle $C$ in the ...
0
votes
1answer
229 views

Shortest distance along the surface of the hyperboloid [closed]

Given two points A and B on the surface of the hyperboloid x^2+y^2-z^2=1. How to find the shortest distance between them along the surface?
5
votes
1answer
199 views

Reference for a dual isoperimetric problem and solution

I am trying to track down the first published solution to the following problem: What curves within the unit disc in the plane and endpoints on the unit circle, minimize their length (within the ...
7
votes
2answers
525 views

Fitting a mesh to a density function

Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). For large $N$, is ...
6
votes
3answers
1k views

Original proof of Pappus' Hexagon Theorem

Does anyone know where I can find an english translation, preferrably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of Pappus' hexagon ...
5
votes
0answers
279 views

degenerating surface II

In degenerating surface, Robert Bryant give us an example of a sequence of minimal immersions which converges (in $C^2$- topology) to $z\mapsto z^{2k+1}$ on the unit disc $\mathbb{D}$. My question is ...
5
votes
1answer
1k views

Good Surface,Bad Surface-Surface classification

Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help. We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by ...
11
votes
2answers
496 views

Periodic lightray paths trapped between two nested mirror circles

I wonder if the periodic paths of a lightray trapped between two nonconcentric circles, each perfectly reflecting, are known? The behavior of such rays seems chaotically complicated. For example, ...
1
vote
1answer
422 views

Program for drawing cobordisms [closed]

Perhaps this is not the right place to ask the following question but I did not find any suitable on the web. So I would be very grateful for sharing your experience. What is a good program to draw ...
4
votes
2answers
958 views

Selecting two random points inside a sphere which are a fixed distance apart

Without appealing to a guess-and-check approach, how might I select a pair of random points inside of a sphere of radius $R$ s.t. the points always a distance $d \leq R$ apart? Can the selected ...
0
votes
1answer
378 views

Minimum distance between two data sets

Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$, $x_{1}\ge\cdots\ge x_{10}>0$ and ...
7
votes
3answers
1k views

Why are isometries of Minkowski space necessarily linear?

The Mazur-Ulam theorem says that any surjective isometry of normed vector spaces is affine. This argument doesn't seem to apply to Minkowski space (of special relativity) since the metric is ...
0
votes
2answers
4k views

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.
6
votes
2answers
768 views

Computational complexity of integration in two dimensions

I have an algorithm for solving a certain problem that requires that I compute two-dimensional integrals as a subroutine, and I'd like to make some kind of statement about its running time. Suppose ...