Questions tagged [curves]

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2
votes
1answer
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
2answers
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
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0answers
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
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0answers
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
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0answers
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
1answer
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
1answer
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
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2answers
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
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0answers
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
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0answers
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
2answers
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
4answers
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
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0answers
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
3answers
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
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1answer
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
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5answers
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
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0answers
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
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0answers
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
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0answers
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
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1answer
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
1answer
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
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0answers
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
1answer
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
1answer
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
1answer
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
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0answers
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
1answer
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
0answers
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
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2answers
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
1answer
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
1answer
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
1answer
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
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0answers
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
0answers
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
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0answers
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
1answer
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
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0answers
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
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1answer
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
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1answer
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
1answer
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
1answer
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
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0answers
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
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2answers
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
3answers
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
1answer
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
2answers
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
0answers
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
1answer
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
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0answers
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
2answers
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. ...