Questions tagged [curves]
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40 questions with no upvoted or accepted answers
19
votes
0
answers
841
views
I found a (probably new) family of real analytic closed Bezier-like curves; is it publishable?
Given $n$ distinct points $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)$ in the plane $\mathbb{R}^2$, I associate a real analytic map:
$f_{\mathbf{x}}: S^1 \to \mathbb{R}^2$
with the following ...
- 5,215
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
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
5
votes
0
answers
413
views
Most divisors on a curve aren't special?
I have a generic smooth curve $C$ of genus $g$ and fixed multiplicities $a_1, \dots, a_n \geq 0$ with $\sum a_i = g+1$.
Q1 : For generic marked points $p_1, \dots, p_n \in C$, must $\sum a_i p_i$ be a ...
- 1,084
5
votes
0
answers
103
views
wild julia sets
Using the Baire category theorem, we may show that most simple closed curves satisfy the following property: any segment between an interior point and an exterior point of the curve intersects the ...
- 18.7k
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
5
votes
0
answers
333
views
Which equation of a Butterfly?
Let $A, B$ be two points and $L$ be a line on the Euclidean Plane. Take two points $J, G$ on the line $L$ such that $JG=constant$. Let $AJ$ meet $BG$ at $P$, $AG$ meet $BJ$ at $Q$, then the locus of ...
- 477
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
5
votes
0
answers
423
views
Cusp point and straightness of a smooth curve.
I have a smooth curve of length $L$ with a single cusp point $P$ occuring at length $s = L_P$. Let the curve in arc length parametrization be $\alpha_t(s) \equiv (X_t(s),Y_t(s)) $. They are actually a ...
- 698
4
votes
0
answers
64
views
$(n-2)$-degree curve passing through $n(n-1)/2$ midpoints
It is known that in the plane, there is an unique conic passing through given $5$ points.
For any $4$ points, there is 6 segments which vertex from these points.
It is known that $6$ midpoints of ...
- 335
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
4
votes
0
answers
122
views
Does there exist curve (for example, in $\mathbb R^2$) that either touches itself or intersects itself at every one of its points?
I really do not even know how to constructively think about this question that I wanted to post before, but delayed. I know that there are space-filling curves and curves of positive area and those ...
user114642
4
votes
0
answers
145
views
How many times do I have to blow up such a curve until it is smooth?
If I have parametrized a curve in the complex plane by $$x(t)=a_lt^l+\cdots+a_nt^n$$
$$y(t)=b_kt^k+\cdots+b_mt^m$$
and the image is reduced (there exist at least two exponents which are relatively ...
- 251
3
votes
0
answers
303
views
An algebraic proof: A line bundle on a curve with a connection must be of degree 0
Let me state it using the language of $D$-modules. Let $X$ be a smooth projective curve and $\mathcal{L}$ a line bundle on it. Assume that $\mathcal{L}$ has a left action of $\mathcal{D}_X$. Then show ...
- 1,168
3
votes
0
answers
160
views
Semistability of restrictions of a semistable vector bundle over a reducible nodal curve
Let $C$ be a reducible nodal curve over complex numbers with two smooth components $C_1$ and $C_2$ intersecting at the only node $P$. Let $E$ be a $\omega$ semistable vector bundle over $C$ of rank $r$...
- 290
3
votes
0
answers
95
views
On the degrees of the normal and tangent sheaves of Gorenstein curves in Pn
It is known that if an irreducible curve $C$ is a local complete intersection in $\mathbb{P}^n$, then
$\wedge^{n-1}\mathcal{N}\otimes\omega_{\mathbb{P}^n}$ is isomomorphic to the dualizing sheaf $\...
3
votes
0
answers
118
views
How order of divisor with support at infinity is changed at reduction?
Jing Yu in his paper "On Arithmetic Of Hyperelliptic Curves" on page 5 asserts the following
The most interesting case is certainly the case $k = \mathbb Q$ and $D \in \mathbb Z[t]$.
To decide ...
- 424
2
votes
0
answers
99
views
What shapes can roll down tailor-made curves, without jumping off or behaving chaotically?
Background
It is well-known that the disk can roll down various curves, including the straight line and the Brachistochrone curve. The latter is the curve along which both a point rolls down the ...
- 4,829
2
votes
0
answers
105
views
Vanishing of Goldman bracket requires simple-closed representative?
Let $\Sigma$ be a connected oriented surface, and $[-,-]\colon \Bbb Z\big[\widehat\pi(\Sigma)\big]\times Z\big[\widehat\pi(\Sigma)\big]\to Z\big[\widehat\pi(\Sigma)\big]$ be the Goldman Bracket. Note ...
- 1,097
2
votes
0
answers
129
views
Single theorem for hybrid of winding number and rotation number?
I am trying to make mathematical sense of some observations from my physics research, so I hope that you will bear with me.
For a complex-valued function $z(t)$ dependent on parameter $t$, I calculate ...
- 121
2
votes
0
answers
50
views
Classification and computation of the entanglements of pairs of planar curves
Let $C$ is the set of continuous $f:[0;1]\to \mathbb R^2$ with $\|f\|=\max_t\|f(t)\|$. For $f,g\in C$ let $(f,g)\in E$ iff $\{f(0),f(1)\}\cap {\rm Range}(g)=\emptyset$ and $\{g(0),g(1)\}\cap {\rm ...
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
1
vote
0
answers
115
views
Cokernel of map of dual of sheaves of differentials/deformations
Let $C$ be a nodal projective curve over an algebraically closed field of genus at least $2$. There are two natural "differential objects" one can consider: The sheaf of differentials $\...
- 223
1
vote
0
answers
80
views
Biot-Savart-like integral for a toroidal helix
The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon.
Let us consider a toroidal helix parametrized as follows:
$$
x=(R+r\cos(n\phi))\cos(\phi)...
- 255
1
vote
0
answers
113
views
Curvature of randomly generated B-spline curve
I am working on Bayesian statistical estimation of parameters (control points) of closed B-spline curve bounding an object on a an image. The problem is that I require those curves to not be much &...
- 31
1
vote
0
answers
119
views
More on points on a curve of genus 3
Let $Y$ be a smooth complex projective curve of genus two,
$X$ a Galois cover of degree two of $Y$ and $K$ the canonical
divisor of $X$. Let $i$ be the involution of $X$ over $Y$.
Can one find two ...
- 237
1
vote
0
answers
50
views
Fibrewise coordinates in a neighborhood of a graph of a continuous curve
Let $M$ be a smooth manifold, $\dim M=n$ and $\gamma:[0;1]\to M$ be continuous. Is it true that there exists local coordinates $(y^0,\ldots,y^n)$ in a neighborhood $V$ of the graph $\{(t,\gamma(t)),t\...
- 61
1
vote
0
answers
34
views
Curves traced out by the centers of mass of rolling convex shapes
Question:
which kind of curves can be traced out by the center of mass of a rigid compact convex shape of uniform density that rolls along the x-axis without slip?
Formulatd differently: are there ...
- 13.2k
1
vote
0
answers
94
views
Constant width curves and inscribed/ circumscribed ellipses
It is known (see for example the Wikipedia entry on the Reuleaux triangle) that for every curve of constant width (CCW), the largest inscribed circle and the smallest circumscribed circle are ...
- 5,979
1
vote
0
answers
75
views
Is there an analytic formula (or even a name...) for a plane curve with curvature inversely proportional to x?
I'm interested in plane curves with curvature inversely proportional to distance from the axis:
$$\kappa(t) = \left(\frac{x'(t) y''(t) - y'(t)x''(t)}{(x'(t)^2 + y'(t)^2)^{3/2}} \right) = \frac{1}{a x(...
1
vote
0
answers
111
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
1
vote
0
answers
239
views
Proposition from Kollar's Rational Curves on Algebraic Varieties
$\DeclareMathOperator\Hom{Hom}$I have some questions on a proof of Proposition II.3.10 & notations from Rational Curves
on Algebraic Varieties by Janos Kollar (page 117).
We work in setting ...
- 5,998
1
vote
0
answers
205
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/...
1
vote
0
answers
52
views
Understanding the Exp map from a moduli of smooth curves
The setup:
Suppose I'm given a family of class $C^k$ curves $\{f_{z}\}_{z\in Z}$ with values in $\mathbb{R}$ parameter by a finite number of parameters $Z\subseteq \mathbb{R}^m$.
Let $\mathscr{M}$ be ...
- 5,405
1
vote
0
answers
60
views
Characterizations of cycloid
There are several constructions of a cycloid. I have some examples below. Are there any others?
Trace of a fixed point on a rolling circle
Evolute of another cycloid (the locus of all its centers of ...
- 761
1
vote
0
answers
516
views
support of embedded points in a curve
Let $C \subset \mathbb{P}^n$ be an one dimensional scheme. Suppose that $C$ decomposes as the union of a Cohen Macaulay reduced curve $\tilde{C}$ (in particular $\tilde{C}$ does not have embedded ...
- 103
0
votes
1
answer
231
views
Divide angles by coefficients relate to Fibonacci sequence
In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle $(...
- 1,776
0
votes
0
answers
66
views
Question on existence of almost length-minimizing curve in a general domain?
I have the following question: for a general domain $\Omega$ in $\mathbb{R}^n$, is it true that for each pair of points $x,y\in \Omega$, there exists a curve $\gamma$ connecting $x$ and $y$ in $\Omega$...
- 11
0
votes
0
answers
116
views
cohomology of curves
Let $X$ be a smooth projective complex curve. Consider the diagonal $\Delta$
in $X \times X$, and $\mathcal{O}(\Delta)$ the associated line bundle.
If $j$ is the inclusion of $\Delta$ in $X \times X$ ...
- 237
0
votes
0
answers
83
views
Dimension of the Representation of the Suzuki and Ree Groups?
What are the dimension of the group representation of $^2B_2$ and $^2G_2$? All what I know is that the first is 4 and the second group has two representation of dimension 7 and 13. Are there any?
- 95