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18
votes
0answers
330 views

The Eyeball Theorem generalized

I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher ...
11
votes
0answers
392 views

Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
10
votes
0answers
341 views

What is the field generated by an Archimedean Spiral?

Let $F \subset \mathbb R$ be the set of lengths of line segments that one can construct, starting from the points $(0,0)$ and $(1,0)$, using a straightedge, compass, and an Archimedean spiral - the ...
9
votes
0answers
749 views

3-piece dissection of square to equilateral triangle?

At a workshop it was suggested that it likely remains an open problem whether or not there is a 3- or 2 -piece dissection of a square to an equilateral triangle. Can anyone confirm that this is ...
8
votes
0answers
115 views

Generalized Hlawka inequality

Let $E$ be a vector space over the real (the complex case is interesting too). We consider functions $f:E\rightarrow\mathbb R$ which satisfy the homogeneity property $$f(\lambda x)=|\lambda|\,f(x).$$ ...
6
votes
0answers
756 views

Interpolating points with minimum curvature constraint

I have $n$ points $p_i$ strictly interior to a rectangle $R$, and I would like to connect them with a curve $C$ whose curvature is as low as possible. Let $\kappa_\max(C)$ be the sharpest (largest ...
3
votes
0answers
58 views

Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...
3
votes
0answers
89 views

Dubins car shortest paths: Decidable?

A Dubins car follows a Dubins path in $\mathbb{R}^2$, with constant wheel speed and limited turning radius. It is known that the shortest Dubins path in the absence of obstacles follows circular arcs ...
3
votes
0answers
42 views

Number of not self-intersecting closed paths spanning $n$ iid uniform points

Let $X_1,X_2,\dots,X_n$ be independent uniform variables in the square. What is the number of piece-wise linear paths which vertices are all the $X_i$ and that do not self-intersect? In other words, ...
3
votes
0answers
133 views

Tetrahedra passing through a hole

Assume a plane $P\subset\mathbb R^3$ has a hole $H$, and that the hole is topologically a compact disc. Being so, $P\setminus H$ does not separate the space. A regular tetrahedron $\sigma^3$ (of ...
3
votes
0answers
418 views

sine and Archimedes' derivation of the area of the circle

The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College ...
3
votes
0answers
195 views

Finite subgroups of the unimodular group

This is related to this MO question (and others as well). Hoping that this will not turn out to be too broad, I would like to know about the 'state of the art' of: 1) The problem of classifying ...
3
votes
0answers
307 views

Characterizations of Euclidean space

I posted this question at math.stackexchange.com but didn't get an answer. Is it a dumb question, eventually? There are three ways of characterizing the abstract Euclidean space $E^n$ that are quite ...
2
votes
0answers
142 views

Probability of 4 Points being in Convex Configuration

Background of my question is, that I would like to implement a parallel preprocessing for a constructing the convex hull of very huge number of points in the euclidean plane; the idea is to process ...
2
votes
0answers
126 views

Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...
2
votes
0answers
188 views

Axiomatization of the incidence geometry of the Euclidean plane

There are several well-known axiomatizations of Euclidean plane geometry, the language of which is usually considered to include at least the relations of incidence (point-line, point-segment, or ...
2
votes
0answers
223 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
vote
0answers
128 views

Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$): $\min_{x\in U}\|x-y\|$. I believe the minimizer $x_{U}^{*}$ is ...
1
vote
0answers
42 views

Components of Intersection of Ellipsoids

Let $\Sigma$ be an $n-1$ dimensional ellipsoid in $\textbf{R}^{n}$ and $S$ the unit sphere. I would like to understand the connected components $C$ of the intersection of $\Sigma$ and $S$. In my ...
1
vote
0answers
95 views

Boundary surfaces in a 3d Voronoi tessellation with obstacles

Let $x_1,\dots,x_n$ be a set of points in $\mathbb{R}^3$ and let $\mathcal{O}_1 ,\dots, \mathcal{O}_m$ denote a set of polyhedral obstacles. What is the name for the surfaces that describe the ...
0
votes
0answers
45 views

Random projections and subspace preservation

Consider a set of points $\{\mathbf{x}_i\}_{i=1}^N$ in a $d$-dimensional linear subspace of $\mathbb{R}^n$ (assume $\{\mathbf{x}_i\}$ span the subspace). The points are projected to a $k$-dimensional ...
0
votes
0answers
90 views

About the area of the region where the paper is twofold when you double a piece of paper in the shape of a triangle

Suppose that you have a piece of paper in the shape of a triangle $ABC$ whose area is $S_0$ and that the area of the region where the paper is twofold when you double the paper in two along a line is ...
0
votes
0answers
88 views

Error Metric which incorporates both mean & standard deviation of data in euclidean space

For simplicities sake (the actually problem is more complex)...Let say I have a set of n 3d points, whose position move over time. For all pairs, I have calculated the mean and standard deviation of ...
0
votes
0answers
156 views

Inertia/Gravity in Distance Geometry

The Cayley-Menger Determinant, D(N), slickly calculates the N-dimensional simplex volume of any N+1 points. One constraint in our 3D world is that D(4)=0. Give each point a mass (Mi) and dynamic ...
0
votes
0answers
278 views

Are XI.1 and XI.7 of Euclid’s Elements equivalent?

Are XI.1 and XI.7 of Euclid’s Elements equivalent? XI.1 A part of a straight line cannot be in the plane of reference and a part in plane more elevated. XI.7 If two straight lines are parallel and ...