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Number of curves in an admissible system of Jordan curves on a surface

Consider a compact Riemann surface of genus $g\geq2$. An admissible system of Jordan curves is a finite collection of Jordan curves $\{\gamma_1,\cdots,\gamma_n\}$ such that they are nonintersecting ...
trisct's user avatar
  • 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 ...
Joe Previdi's user avatar
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 $X \times X$ ...
user95246's user avatar
  • 237
2 votes
1 answer
300 views

Smooth transformation of a curve with fixed ends and length [duplicate]

I am simulating polymers of fixed length and fixed ends. I would like to search the phase space of all possible conformations quickly. Is there anyway I can generate efficiently a lot of (rather) ...
Davood Norouzi's user avatar
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 ...
coudy's user avatar
  • 18.7k
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}{...
Manfred Weis's user avatar
  • 13.2k
1 vote
1 answer
430 views

Do negative indecomposable bundles on curves have sections?

Let $X$ be a smooth projective curve, and $E$ an indecomposable vector bundle on $X$ with $\mathrm{deg} E<0$. Is it true that $H^0(X,E)=0$? This is true if $E$ is a line bundle, which means it is ...
Nathan's user avatar
  • 39
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 \...
Jiří Minarčík's user avatar
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 $\textit{F}$ ...
Leonardo's user avatar
  • 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{...
Jiří Minarčík's user avatar
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 $\...
Andre Contiero's user avatar
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/...
Niculae George Razvan's user avatar
7 votes
1 answer
676 views

A problem of four conics

I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof: Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...
Đào Thanh Oai's user avatar
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 ...
Leonardo's user avatar
  • 405
4 votes
1 answer
265 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?
Minghui Ouyang's user avatar
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 $\...
Ali Taghavi's user avatar
8 votes
1 answer
441 views

Axioms of length

Assume I want to define length of plane curves axiomatically. It seems to be reasonable to assume that The length of a unit segment is 1; Congruent curves have equal lengths; Length is additive with ...
Anton Petrunin's user avatar
6 votes
1 answer
766 views

Rationality of the moduli space of genus g curves

I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions ...
skd's user avatar
  • 5,760
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 ...
David Eppstein's user avatar
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 ...
Niven Zhao's user avatar
5 votes
3 answers
479 views

Is there a dense subset on closed Jordan curve $C$ which its points make intersections under certain rotations?

Is it true that for any given closed Jordan curve of $C \subset \mathbb{R}^2$ there is a dense subset $A$ such that for every point $p\in A$ we have the following property: If we rotate $C$ around $p$...
MasM's user avatar
  • 289
0 votes
1 answer
61 views

Is the difference between polyfits for two data series equivalent to the polyfit of the difference between the two data series?

Suppose that we have two series of data points $a(x)$ and $b(x)$ with the same domain of definition for $x$, and we fit two polynomial functions $f(x)$ and $h(x)$ (of the same order $n$) to them, ...
nino's user avatar
  • 147
8 votes
2 answers
587 views

Jordan curves in $\mathbb R^n$ and inscribed equilateral triangles

Inscribed square problem wants that we know "Does every Jordan curve admit an inscribed square?" From my amateur viewpoint it seems that the concept of Jordan curve can be straightforwardly ...
user avatar
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 ...
user avatar
2 votes
1 answer
123 views

Segments on a closed convex plane curve

Is it true that there does not exist a closed convex plane curve containing an infinite number of segments, belonging to distinct lines each?
Evgenii.Balai's user avatar
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 ...
Taylor's user avatar
  • 251
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. ...
David G. Stork's user avatar
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. ...
coudy's user avatar
  • 18.7k
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 ...
Cố Gắng Lên's user avatar
4 votes
1 answer
312 views

A Geometric Combinatorial/Graph Theory Question

I have a combinatorics problem that seems pretty general - I'd be surprised if the answer is not known. Unfortunately, I can't seem to solve it. The question concerns the following situation: ...
John Samples's user avatar
7 votes
4 answers
4k views

Generating random curves with fixed length and endpoint distance

Are algorithms already known, that generate (arbitrarily good approximations of) random curves, w.l.o.g. with unit length, and joining endpoints $(0,0)$ and $(\alpha,0)$ with $\alpha \lt1$ given? The ...
Manfred Weis's user avatar
  • 13.2k
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 ...
Ivo Terek's user avatar
  • 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 ...
Joseph O'Rourke's user avatar
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 ...
Ivo Terek's user avatar
  • 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 ...
poupy's user avatar
  • 175
9 votes
1 answer
1k views

A chain of six circles associated with a conic

I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems: A chain of six circles ...
Oai Thanh Đào's user avatar
4 votes
1 answer
684 views

A problem of four curves

This is a generalization of my previous question, a problem of a cubic and six conics. Let a curve $(K)$ of degree $m$ and three curves $(C_i)$ of degree $n$, for $i=1,2,3$. Let $(C_1)$ meets $(K)$ ...
Oai Thanh Đào's user avatar
10 votes
3 answers
894 views

Automorphisms of cartesian products of curves

Let $C$ be a smooth projective curve. Is it true that $$\textrm{Aut}(C\times C)\cong S_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$ and in case, what would be a reference for this? Thanks.
user avatar
6 votes
1 answer
321 views

Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)

In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid. Could you ...
Bear's user avatar
  • 845
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 ...
Manfred Weis's user avatar
  • 13.2k
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 ...
Maxim's user avatar
  • 424
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 ...
ABIM's user avatar
  • 5,405
3 votes
1 answer
191 views

Second symmetric product of a hyperelliptic curve

Let $C$ be a hyperelliptic curve of genus $g\geq 3$, let $C^{(2)}$ be the second symmetric product of $C$ with itself, i.e. the quotient of $C\times C$ by the involution $(p,q)\mapsto (q,p)$ and let $...
antonio's user avatar
  • 31
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 $r>\frac12$ (...
Wolfgang's user avatar
  • 13.4k
2 votes
1 answer
182 views

Parametric smooth curve that vists all integer points of the plane [closed]

Does there exist a parametric smooth curve that visits all integer points $(x,y),\, x,y \in \mathbb{N}$ of the plane? Something similar to this: $$\begin{align} x = &\theta \cos(2\sin(\theta\pi))...
Marzio De Biasi's user avatar
4 votes
1 answer
226 views

A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem

I propose a conjecture variant of Cayley-Bacharach's theorem. I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know ...
Oai Thanh Đào's user avatar
4 votes
1 answer
578 views

A conjecture like Cayley–Bacharach theorem

Let six points $A, A', B, B', C, C'$ lie on a conic and a cubic. Let a conic through $B, B', C, C'$ and meets the cubic again at $A_1, A_2$. Let a conic through $C, C', A, A'$ and meets the cubic ...
Oai Thanh Đào's user avatar
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 ...
Erfan Salavati's user avatar
7 votes
1 answer
899 views

Isotrivial families with non-zero Kodaira spencer map

Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this ...
Pancho's user avatar
  • 171
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 ...
Rajesh D's user avatar
  • 698