# Questions tagged [curves]

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78
questions

**18**

votes

**0**answers

600 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 ...

**0**

votes

**1**answer

160 views

### property of rational functions on projective curves

I have a couple of question about the proof of Lemma 1.20.5 from Janos Kolloar's Lecture on Resolution of Singularities (page 19):
Lemma 1.20.5 Let $C$ be a reduced, irreducible projective curve (=...

**2**

votes

**1**answer

302 views

### Very weak Riemann-Roch on curves (by J. Kollar)

I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):
1.13 (Very weak Riemann-Roch on ...

**1**

vote

**0**answers

161 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 ...

**3**

votes

**1**answer

99 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 ...

**3**

votes

**1**answer

70 views

### 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 ...

**3**

votes

**1**answer

92 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 ...

**0**

votes

**0**answers

95 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 $X \times X$ ...

**2**

votes

**1**answer

85 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) ...

**5**

votes

**0**answers

90 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 ...

**0**

votes

**0**answers

42 views

### Family of space curves with ratio torsion/curvature changing linearly

I was wondering what happens if we take a closed space curve and modify its curvature and torsion in a way their ratio changes linearly. Formally:
Let $\gamma(s):[0,2\pi]\rightarrow \mathbb{R}^3$ be ...

**1**

vote

**2**answers

111 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}{...

**1**

vote

**1**answer

218 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 ...

**2**

votes

**1**answer

98 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 \...

**4**

votes

**1**answer

136 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 $\textit{F}$ ...

**7**

votes

**0**answers

132 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{...

**3**

votes

**0**answers

65 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 $\...

**1**

vote

**0**answers

131 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

**1**answer

241 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 ...

**5**

votes

**0**answers

158 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 ...

**4**

votes

**1**answer

148 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?

**3**

votes

**1**answer

209 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 $\...

**8**

votes

**1**answer

290 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 ...

**6**

votes

**1**answer

348 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 ...

**13**

votes

**0**answers

211 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 ...

**9**

votes

**2**answers

974 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 ...

**5**

votes

**3**answers

386 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 $p$...

**0**

votes

**1**answer

54 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, ...

**7**

votes

**2**answers

321 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 ...

**4**

votes

**0**answers

110 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 ...

**1**

vote

**1**answer

98 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?

**4**

votes

**0**answers

130 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 ...

**19**

votes

**1**answer

442 views

### Jordan curves admitting only acyclic inscriptions of squares

The (unsolved) inscribed square problem or Toeplitz conjecture posits that every closed, plain continuous (Jordan) curve ${\it \Gamma}$ in $\mathbb{R}^2$ contains all vertices of some square. Neither ...

**7**

votes

**1**answer

217 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.
...

**5**

votes

**0**answers

311 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 ...

**4**

votes

**1**answer

257 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: ...

**7**

votes

**4**answers

1k 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?
...

**2**

votes

**1**answer

155 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 ...

**15**

votes

**1**answer

519 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 ...

**5**

votes

**0**answers

290 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 ...

**2**

votes

**1**answer

505 views

### Relation between Cox-deBoor recursion and Convolution (b-spline basis)

Consider the Cox-deBoor recursion formula for producing b-spline basis functions given a knot vector:
$N_{i,0}(u)=1 $ if $u_i\leq u < u_{i+1}$
otherwise, $=0$
$N_{i,p}(u)=\frac{u-u_{i}}{u_{...

**1**

vote

**1**answer

55 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 ...

**9**

votes

**1**answer

833 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 ...

**4**

votes

**1**answer

647 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 $(K)$ ...

**11**

votes

**3**answers

692 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.

**6**

votes

**1**answer

205 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 ...

**2**

votes

**0**answers

33 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 ...

**3**

votes

**0**answers

102 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 ...

**1**

vote

**0**answers

51 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 ...

**3**

votes

**1**answer

160 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 $...