# Questions tagged [curves]

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95
questions

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votes

**1**answer

71 views

### Hemispherical space filling hilbert curve

First question here, sorry for any posting infractions.
I need to create/find/buy a hemispherical space-filling Hilbert(or similar) curve.
something similar to Cube hilbert
but only filling a ...

**5**

votes

**2**answers

307 views

### Grand tour of the special orthogonal group

Is there a continuous function $f:[0,+\infty) \to \operatorname{SO}(n)$ whose image is dense in $\operatorname{SO}(n)$ and that is well behaved in certain ways?
For each $\varepsilon>0$ it doesn't ...

**2**

votes

**0**answers

76 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 $r$...

**1**

vote

**0**answers

56 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

74 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, $t$, $n$ ...

**5**

votes

**1**answer

272 views

### Higher order inflection points

Consider a smooth plane curve $X\subset\mathbb{P}^2$ of degree $d$. We will say that $x\in X$ is an inflection point of order $s$ if the tangent line $T_xX$, of $X$ at $x\in X$, intersects $X$ in $x\...

**3**

votes

**1**answer

143 views

### Configuration of points on a plane curve

Let $C\subset\mathbb{P}^2$ be a smooth plane curve of degree six. On $C$ there are $21$ points given as the intersection points of two lines choosen among a set of seven lines. More precisely there ...

**11**

votes

**2**answers

229 views

### Connecting a compact subset by a simple curve

Let $K$ be a compact subset of $\mathbb R^n$ with $n\ge 2$ (say if you like $n=2$, which is possibly sufficiently representative).
Q: Does there exist a closed simple curve $u:\mathbb S^1\to\mathbb R^...

**5**

votes

**0**answers

272 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 ...

**0**

votes

**0**answers

53 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 $\Omega$...

**6**

votes

**2**answers

320 views

### Snake algorithm that minimizes straight lines

How can I find the non-self-intersecting loop that uses the least amount of straight lines (curves left/right as often as possible every turn) and still loops back on itself?
Here's an example we have ...

**70**

votes

**4**answers

2k views

### Does this property characterize straight lines in the plane?

Take a plane curve $\gamma$ and a disc 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 $\...

**1**

vote

**0**answers

40 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 ...

**23**

votes

**3**answers

3k views

### Cardioid-looking curve, does it have a name?

The curve, given in polar coordinates as $r(\theta)=\sin(\theta)/\theta$
is plotted below.
This is similar to the classical cardioid, but it is not the same curve (the curve above is not even ...

**1**

vote

**1**answer

278 views

### Is this curve well known?

I consider the curve $c(t)=(x(t),y(t))$ in $\mathbb{R}^2$ such that
$\frac{d^2x(t)}{dt^2}=-(a\sin t+b)\frac{dy(t)}{dt}$
$\frac{d^2y(t)}{dt^2}=(a\sin t+b)\frac{dx(t)}{dt}$
$a,b\in\mathbb{R}$
Is the ...

**25**

votes

**5**answers

1k views

### Surprising properties of closed planar curves

In https://arxiv.org/abs/2002.05422 I proved with elementary topological methods that a smooth planar curve with total turning number a non-zero integer multiple of $2\pi$ (the tangent fully turns a ...

**0**

votes

**0**answers

81 views

### What can we say about the complement of two homotopic simple closed curves on a compact orientable surface?

Do two homotopic simple closed curves separate a compact orientable surface?
If they are disjoint they do and bound a cylinder. But if they intersect? What can we say about the components of their ...

**4**

votes

**0**answers

67 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 ...

**19**

votes

**0**answers

754 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 ...

**-1**

votes

**1**answer

177 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 (=1D ...

**2**

votes

**1**answer

323 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 curves)...

**1**

vote

**0**answers

179 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

71 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

119 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

98 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

121 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

95 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 ...

**1**

vote

**2**answers

114 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

243 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

106 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

143 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

163 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

69 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

153 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/...

**6**

votes

**1**answer

429 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

182 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

162 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

215 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

339 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

403 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 ...

**14**

votes

**0**answers

234 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 ...

**11**

votes

**2**answers

1k 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

403 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

57 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

382 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 ...

**2**

votes

**1**answer

108 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

136 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 ...

**22**

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. ...