All Questions
Tagged with euclidean-geometry dg.differential-geometry
15
questions
6
votes
1
answer
165
views
$\mathbb{Q}$-rank of the space of angles of pythagorean triples
A pythagorean triple is a triple of integers $(a,b,c)$ with $a^2 + b^2 = c^2$. Given a triple, $(a/c, b/c)$ is a point on the unit circle, so we may associate to it the normalized angle
$$\theta_{a,b} ...
- 7,313
19
votes
1
answer
815
views
All saddles in the unit ball have area $<2\pi$?
Let $M$ be the saddle surface in $\mathbb R^3$ defined by $x^2-y^2+z=0$. For any $r\geq 0$ and $(x_0,y_0,z_0)\in\mathbb R^3$, let $rM+(x_0,y_0,z_0)$ denotes the surface obtained by scaling $M$ by $r$ ...
- 513
1
vote
2
answers
309
views
Is there an area-preserving concentric diffeomorphism of the ellipse?
$\DeclareMathOperator\Vol{Vol}$This is a cross-post.
Let $0<b<1$ be a fixed parameter, and let $(R(\theta),\theta)$ be the polar coordinates of the ellipse
$$E=\{(x,y) \in \mathbb R^2 \, | \, \...
- 6,641
1
vote
0
answers
93
views
Differential of the gradient of a strictly convex function
For $n\geq 2$, we consider $\mathbb{R}^n$ endowed with the usual scalar product. Let $f\in\mathcal{C}^2(\mathbb{R}^n,\mathbb{R})$ be a striclty convex function such that $\nabla f$ is nowhere ...
- 629
90
votes
5
answers
4k
views
Does this property characterize straight lines in the plane?
Take a plane curve $\gamma$ and a disk of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\...
- 4,341
1
vote
1
answer
229
views
On the crookedness of curves (Milnor's paper)
I am reading the paper (Ann Math. 1950) "on the total curvature of knots" by J. Milnor.
I was trying to understand this part of the paper (here is a free access link to the paper):
Let $P$ be a ...
- 231
4
votes
1
answer
157
views
What curve of positive curvature minimizes distance from the origin, given length and total curvature?
Let $\textit{F}$ be the family of $C^1$ curves in $\mathbb{R}^2$ of fixed length $\bar{l}$ and fixed tangent's turning angle $\bar{k}$.
What are the curves of positive curvature in $\textit{F}$ ...
- 405
80
votes
1
answer
3k
views
Converse to Euclid's fifth postulate
There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, ...
- 7,087
8
votes
1
answer
265
views
Isoperimetric inequality on the plane
Let $A$ be a connected compact domain with smooth boundary in the Euclidean 2-plane. Assume its diameter is at most $d$. Assume that the second fundamental form of the boundary is at most $-c$ where $...
- 21.3k
0
votes
1
answer
121
views
Triangle inside the Closed Curve
For any piece wise smooth, simple closed curve $\gamma$ in the Euclidean plane $E^2$ and fix a point $G$ inside the area circled by $\gamma$.
Show: There exists three points $A,B$ and $C$ on the $\...
- 1,915
7
votes
2
answers
946
views
Explicitly describing the region of the plane "outward of" a simple, open, oriented, cubic curve $c:(0,1)\to\mathbb{R}^2$
Some Context:
I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an ...
- 279
12
votes
1
answer
1k
views
How large can you draw an island on a map?
A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...
- 1,337
8
votes
1
answer
1k
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 ...
- 150k
3
votes
1
answer
161
views
Symmetry group for the frame bundle of a G-space
Let $Q$ be a smooth manifold, and let $G$ be a Lie group which acts smoothly on $Q$ on the left.
Question 1: does the group $G$ act naturally on the tangent bundle $TQ \to Q$?
My motivation here is ...
- 8,422
13
votes
2
answers
658
views
Helix translates as geodesics
I believe one can fill $\mathbb{R}^3$ with
horizontal translates of the helix
$(\cos t, \sin t, t) \;,\; t \in \mathbb{R}$,
so that every point of $\mathbb{R}^3$
lies in exactly one helix.
I am ...
- 150k