All Questions
Tagged with curves dg.differential-geometry
29 questions
3
votes
3
answers
457
views
Difference in length of two dimensional concentric closed paths
Two bicyclists ride side by side around a smooth loop with the outside bicyclist a distance of $D$ from the inside bicyclist.
How much further does the outside bicyclist ride?
If the loop is a circle, ...
- 31
1
vote
0
answers
113
views
Curvature of randomly generated B-spline curve
I am working on Bayesian statistical estimation of parameters (control points) of closed B-spline curve bounding an object on a an image. The problem is that I require those curves to not be much &...
- 31
1
vote
0
answers
34
views
Curves traced out by the centers of mass of rolling convex shapes
Question:
which kind of curves can be traced out by the center of mass of a rigid compact convex shape of uniform density that rolls along the x-axis without slip?
Formulatd differently: are there ...
- 13.2k
1
vote
0
answers
94
views
Constant width curves and inscribed/ circumscribed ellipses
It is known (see for example the Wikipedia entry on the Reuleaux triangle) that for every curve of constant width (CCW), the largest inscribed circle and the smallest circumscribed circle are ...
- 5,979
1
vote
0
answers
112
views
Existence theory for geometric flow of space curves
Is there any existence theory applicable to general geometric flows of space curves in the following form?
$$
\partial_t \gamma = v_t t + v_n n + v_b b
$$
Here $\gamma$ is the evolving curve, $t$, $n$ ...
- 11
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 $\...
- 4,459
4
votes
0
answers
101
views
Closed curves with minimal total curvature in the unit circle
Chakerian proved in this paper that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L.
In this later paper Chakerian gave a simpler proof and noted that ...
- 4,862
3
votes
1
answer
125
views
Can a closed horizontal trajectory on a Riemann surface be freely homotopic to $0$?
Let $R$ be a Riemann surface and let $\varphi=\varphi(z)dz^2$ be a nonzero holomorphic quadratic differential on $R$. A differentiable curve $\gamma$ on $R$ is called a horizontal trajectory if along ...
- 283
3
votes
1
answer
234
views
Large class of curves which only intersect each other finitely many times
I am trying to find a large subset of piecewise-differentiable plane curves of finite length (subsets of $\mathbb{R}^2$) with the following property:
For any pair $\gamma_1, \gamma_2$ of curves in ...
- 509
1
vote
2
answers
145
views
Envelope of Ellipses with Common Major-axis Length
are the envelopes of the following set of ellipses "classical" curves of mathematics, that appear naturally as the solution of a mathematical/physical problem;
$$\mathcal{E}:=\Biggr\{\frac{(x-e)^2}{...
- 13.2k
2
votes
1
answer
135
views
Closest points of curves on convex surfaces
Let $\Sigma$ be a convex surface with strictly positive curvature in $\mathbb{R}^3$ and $\Gamma \subset \Sigma$ be a closed simple space curve with parameterization $\gamma : \mathbb{S}^1 \rightarrow \...
- 151
4
votes
1
answer
160
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
9
votes
0
answers
244
views
Interesting geometric flow of space curves with non-vanishing torsion
Recently, while thinking about CMC surfaces, I came up with an interesting geometric flow for curves in $\mathbb{R}^3$ given by
\begin{equation}
\partial_t \gamma = \tau^{-\frac{1}{2}} n,
\end{...
- 151
1
vote
0
answers
206
views
Find wrapping angle of helix on a torus
I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus.
The wrapping angle (or the angle measured around and/...
5
votes
0
answers
237
views
Explicit parametrization of closed space curves of constant curvature
Joining arcs of helices it is pretty easy to obtain closed curves with constant curvature and $C^2$ regularity (see http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0203.pdf). Joining arcs of Salkowski ...
- 405
4
votes
1
answer
266
views
A variation on four-vertex theorem
Is it true that, if a closed, strictly convex curve has exactly four vertices (extrema of curvature), then any circle has at most four points of intersection with it?
- 247
3
votes
1
answer
224
views
Can we foliate the space $\mathbb{R}^3$ with Frenet curves whose tangent and normal vectores span a given $2$ dimensional distribution?
Let $D$ be a $2$ dimensional distribution of $\mathbb{R}^3$.
Is there a $1$ dimensional foliation of $\mathbb{R}^3$ with Frenet curves such that for every leaf $\gamma$ of the foliation we have $\...
- 356
15
votes
0
answers
330
views
How much smoothness does the tennis ball theorem need?
The tennis ball theorem states that a smooth-enough curve that bisects the surface area of a sphere must have at least four inflection points. There are plenty of sources on this but most of them are ...
- 18.6k
12
votes
2
answers
2k
views
A necessary and sufficient condition for a space curve to lie on a ellipsoid
Any (arc-length parametrized) space curve is uniquely determined (up to rigid motion) by its curvature and its torsion.
For instance we know that a necessary and sufficient condition for a space ...
- 121
23
votes
2
answers
1k
views
Jordan curves admitting only acyclic inscriptions of squares
The (recently solved) inscribed square problem or Toeplitz conjecture posits that every closed, plane continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all vertices of some square. ...
- 2,558
7
votes
1
answer
336
views
Uniformisation for non simple closed curves
Given a simple closed curve in the plane, there is a homeomorphism from the unit open disk to the interior of the curve. The homeomorphism can be taken conformal, this is the uniformisation theorem.
...
- 18.7k
2
votes
1
answer
184
views
Relation of pseudo-torsion with curvature in degenerate plane
Question: I'd like to know if there is some reference or reasonable way to develop curve theory in a plane with degenerate metric $(\Bbb R^2, {\rm d}s^2 ={\rm d}x^2)$.
Context: In Lorentz-Minkowski ...
- 1,163
16
votes
1
answer
667
views
Can a shape rolling inside itself reproduce that shape?
Q. Is the circle the only shape that, when rolling inside itself,
has a point that draws out a scaled copy of itself?
Let $C$ be a simple, closed, smooth curve in the plane.
(Likely "smooth" can be ...
- 151k
5
votes
0
answers
327
views
"Correct" definition of signed curvature in Minkowski plane
We know that for $n\geq 2$ the de Sitter space $\mathbb{S}^n_1(r)$ and the hyperbolic space $\mathbb{H}^n(r)$ have constant curvature $1/r^2$ and $-1/r^2$, respectively.
Looking at references such as ...
- 1,163
1
vote
1
answer
64
views
Unbounded convex domains in 2D
Let $\gamma$ be a smooth planar curve. Assume that $\gamma$ divides the plane into two domains and, it addition, that one of these domains is unbounded and convex. What can be said about the behavior ...
- 175
2
votes
0
answers
34
views
Least Width of Planar Unimodal Curves with Unit Diameter
I am currently trying to find a way to define some notion of "roundness" for subtours in graphs and that definition should only be based on the comparing (sums of) edge length and on the order in ...
- 13.2k
10
votes
1
answer
896
views
A tricky tractrix question about vertical tangents
This is raised by a recent question occurring in combinatorial geometry.
It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ (...
- 13.4k
7
votes
2
answers
1k
views
radius of tubular neighborhood
Hi there,
Is there any result about the calculation of radius of tubular neighborhood of submanifold inside a Riemannian manifold?
For example, given a simple smooth curve on R^2, what's the radius ...
- 583
10
votes
4
answers
7k
views
How can I find the average of two 2D curves?
I have a curve interpolation problem.
I have two closed curves that are defined on an X,Y plane. How can I define a 3rd curve that is the average of those two? Programmatically, I have a list of ...
- 203