Deprecated; please use a more specific tag.

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18 views

### minimizing squared distance for point to a set of lines [migrated]

I have this problem that I cannot figure out how to solve. It is from Szeliski's computer vision book (http://szeliski.org/Book/drafts/SzeliskiBook_20100903_draft.pdf) p.94 (electronic version) and is ...

**0**

votes

**0**answers

20 views

### Geometry - How do I find this angle in terms of length/radius [migrated]

I hope I can be clear in my request, english is not my first language.
First, take a look at this simpler diagram: only two arcs
Here is a link to my picture, need more rep apparently.
...

**-1**

votes

**0**answers

5 views

### get points out of two lines where y distance is given value [migrated]

there are two lines form 3 point like so
Point 1 (1,2)
Point 2 (5,6)
Point 3 (7,1)
Lines are from
1 to 2
and from
1 to 3
what i need are the points on each line where the ...

**6**

votes

**3**answers

843 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 ...

**16**

votes

**2**answers

871 views

### Can the unsolvability of quintics be seen in the geometry of the icosahedron?

Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...

**1**

vote

**2**answers

4k views

### 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 ...

**5**

votes

**1**answer

251 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 ...

**1**

vote

**1**answer

129 views

### Determining A valid Convex Hexagon given The length of Six sides

Suppose We are given the length of all six sides of a Convex Hexagon. How can we tell whether it's valid or Not ? that means can we tell whether it's area is positive or not ?

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vote

**0**answers

134 views

### To what extent can Gielis' so-called “Superformula” be used to improve the efficiency of WIFI antennas? [closed]

In 2003, the Belgian plant biotechnologist Johan Gielis proposed a formula that allows for the description of a wide variety of shapes in 2$d$, 3$d$ and higher dimensions. This is the formula
$$r( ...

**14**

votes

**2**answers

242 views

### Random rings linked into one component?

Let $S$ be a sphere of unit radius.
Let $C_n$ be a collection of unit-radius circles/rings whose centers
are (uniformly distributed)
random points in $S$, and which are oriented (tilted) randomly ...

**6**

votes

**5**answers

557 views

### covering by spherical caps

Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius ...

**2**

votes

**2**answers

239 views

### The probability distribution for vertex degree in a unit disc graph generated from random points on a plane

Imagine I cover an arbitrarily large plane with randomly placed points at some density $\rho$ s.t. the number of points in any randomly sampled area $A$ (of arbitrary shape and size) is $\approx ...

**14**

votes

**4**answers

680 views

### Metrics for lines in $\mathbb{R}^3$?

I seek a metric $d(\cdot,\cdot)$ between pairs of (infinite) lines in $\mathbb{R}^3$.
Let $s$ be the minimum distance between a pair of lines $L_1$ and $L_2$.
Ideally I would like these properties:
...

**3**

votes

**1**answer

377 views

### How to partition a graph into N groups with M elements nearest?

Hi.
This problem probably already here but I could not find the right words to find it.
I have a list with 1700 points (geographic coordinates) and a need to separate into 17 groups with 100 ...

**8**

votes

**2**answers

632 views

### Will a ball fired through a focus of an ellipse eventually tend to a horizontal line?

A couple of years ago I came across this phenomenon which appears to be true although I am having difficulty proving it.
F and F' are foci of a billiard table in the shape of an ellipse. A ball is ...

**28**

votes

**6**answers

2k views

### Is it possible to partition $\mathbb R^3$ into unit circles?

Is it possible to partition $\mathbb R^3$ into unit circles?

**6**

votes

**1**answer

413 views

### What is the shape of the $n$-gon which gives the maximum of a function?

What is the shape of the $n$-gon $P_1P_2\cdots P_n$ which gives the maximum of $A_n$? The quantity $A_n$ is defined by
$$ A_n = \frac{{\sum_{i\lt{j}\le{n}}{\lvert P_i ...

**3**

votes

**0**answers

2k views

### What is the geometric meaning of third derivative of a function at a point? [closed]

What is the geometrical meaning of third derivative of a function at a point?
The question is now open in the sister site ...

**2**

votes

**1**answer

221 views

### What is the shape of the convex $n$ -gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $B_n$? ...

**3**

votes

**1**answer

249 views

### Holomorphic coordinates on a Kähler manifold

Let $(X,J,\omega)$ be a Kähler manifold. Let $\dim_{\mathbb{R}}(X)=2n$ and we also know that $X$ splits as $X=M\times \mathbb{R}$, where $\dim_{\mathbb{M}}=2n-1$. My question is now: does there exists ...

**4**

votes

**0**answers

234 views

### Averaging lengths and distances

A natural way in which Finsler metrics appear in Riemannian geometry is as averages of Riemannian metrics (e.g., the average of the arc-length elements
$\sqrt{dx^2 + dy^2}$ and $\sqrt{2dx^2 + 3dy^2}$ ...

**14**

votes

**2**answers

374 views

### Repeated random two-steps in $\mathbb{R}^3$: unbounded?

I created a random isometry $T$ of $\mathbb{R}^3$ by generating
a random orthogonal matrix $M$,
uniformly distributed among all such,
and a random displacement $v$, whose coordinates
are drawn from a ...

**3**

votes

**1**answer

180 views

### cover and hide with squares

I am studying two numbers, related to squares, that can characterize a polygon P:
MinCoverNumber = the minimum number of axis-aligned squares required to exactly cover P (the covering squares may ...

**21**

votes

**5**answers

1k views

### What is a good method to find random points on the n-sphere when n is large?

As part of a more complex algorithm, I need a fast method to find random points of the n-sphere, $S^n$, starting with a RNG (random number generator). A simple way to do this (in low dimensions at ...

**20**

votes

**2**answers

1k views

### Functions whose gradient-descent paths are geodesics

Let $f(x,y)$ define a surface $S$
in $\mathbb{R}^3$ with a unique local minimum at $b \in S$.
Suppose gradient descent from any start point $a \in S$
follows a geodesic on $S$ from $a$ to $b$.
(Q1.)
...

**2**

votes

**0**answers

84 views

### Reachability in dynamic random graphs

There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...

**14**

votes

**5**answers

601 views

### Historical use of figures in geometry

I was surprised to learn from John Stillwell's comment in answer to the
question,
"Can the unsolvability of quintics be seen in the geometry of the icosahedron?",
that
There is not a single ...

**4**

votes

**1**answer

450 views

### Any reference to an algorithm for finding the largest empty circle on a sphere (with great-circle distance)?

Given a set $S$ of 2D points in the plane there are known algorithms for finding the largest empty circle ($LEC$) of the set of points.
The $LEC$ problem is stated in this way: find a $LEC$ whose ...

**5**

votes

**1**answer

147 views

### Matching on sphere to create cycle with chords

Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through
the center of $S$, in such a way that no pair of chords intersect:
I would like ...

**0**

votes

**0**answers

98 views

### Area on the unit sphere swept out by big circles corresponding to a curve

For a point on the unit sphere, we know the plane perpendicular to the line through the origin and the point cuts the sphere with a big circle. When the point moves along a sphere curve, the ...

**8**

votes

**2**answers

795 views

### Non-Kahler Complex manifolds

For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define ...

**3**

votes

**2**answers

262 views

### Symmetric convex curve

a) Assume that $\gamma$ is a symmetric convex curve w.r.t. two orthogonal lines (such a curve is the ellipse). Is the following statement true. There exists a $(l,L)$-bi-Lipschitz mapping of the unit ...

**3**

votes

**1**answer

112 views

### Existence of Simple Closed Straightest Geodesics

There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov].
Is the same true ...

**9**

votes

**0**answers

177 views

### Covering of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes,
that is, it is a curve of constant width.
(The width between two parallel tangents to the curve are independent of the ...

**20**

votes

**5**answers

2k views

### Nonconvex manhole covers

One common reason given for the circularity of manhole covers is so they can't fall through the manhole. For convex manhole covers, this property is equivalent to having constant width -- if you have ...

**1**

vote

**1**answer

137 views

### Construction of an integral point set given the set of distances ,its minimal description so as to get a measure of its complexity and its unique identifier.

Given a set of distances between every pair of points of an integral point set P of n points; say
D = {${d_i}$}
Q1. What is the least time complexity
possible/known for recreating the
...

**7**

votes

**0**answers

254 views

### Expanding disks lead to what packing of the plane?

Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows ...

**2**

votes

**1**answer

62 views

### Covering the annulus of d-cube

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C+x$ = {$~\lambda c+x|c\in C$} for some $x\in R^d$ is called homothetic copy of $C$. The number
...

**1**

vote

**1**answer

148 views

### Helly's number from biconvex functions

Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...

**0**

votes

**0**answers

74 views

### Formal definitions for a few lattice packing invariants

I'm a bit hesitant to post this question here because it regards definitions that are likely trivial, and please note that I'm cross-posting this question from Math.Stackexchange (posted nine days ...

**0**

votes

**0**answers

36 views

### Covering the annulus of symmetric convex body

Consider a symmetric convex body $A$ in $R^d$. Now, we draw another object, $A'$, concentric and translated with respect to A and having radius slightly greater than twice to the radius of $A$.
Now ...

**1**

vote

**1**answer

87 views

### Riemann isometry vs Euclidean bi-Lipschitz mapping

Assume that $\gamma$ is a rectifiable Jordan curve in the complex plane of length $2\pi$. Then there exists a Riemann isometry $f$ between $\gamma$ and the unit circle $T$. My question is, does this ...

**-3**

votes

**1**answer

2k views

### How to transform a plane into a sphere? [SOLVED] [closed]

Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into ...

**4**

votes

**1**answer

175 views

### Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the ...

**7**

votes

**0**answers

327 views

### Rectangology and squareology

I thought that rectangles were simple, and squares even simpler. Until my research has led me to several questions about rectangles and squares, which I can't solve.
I started by posting this ...

**1**

vote

**2**answers

710 views

### Trilateration problem

When trying to develop an algorithm for a program, I got with the following problem:
Determine the approximate location of $O$, if you can take finite samples $P_n$ from known locations and always ...

**2**

votes

**1**answer

137 views

### Helly's Theorem for Biconvex Sets

Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...

**3**

votes

**1**answer

179 views

### convex polyhedron in the unit cube

Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $conv(Q) \supseteq P$ where $conv(Q)$ ...

**9**

votes

**1**answer

298 views

### Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...

**2**

votes

**0**answers

109 views

### Blowing up spheres in a face centered cubic (fcc) packing geometry just enough to cover the volume of the lattice

Imagine I have an infinite lattice of spheres packed in a face centered cubic (fcc) lattice geometry which has the basis: $((-1, -1, 0), (1, -1, 0), (0, 1, -1))$. Here, provided that sphere-sphere ...