Questions tagged [curves]

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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 ...
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3 votes
1 answer
96 views

Maps that preserve winding numbers

This question is a cross post from the Math StackExchange since it got no attention at all there: https://math.stackexchange.com/questions/4414601/maps-that-preserve-winding-numbers I am looking for a ...
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0 answers
111 views

why Alexander method gives us a finite combinatorial problem?

The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups" For a surface $S$ with marked points, we say that a collection $\left\{\gamma_{i}\right\}$ of curves and arcs ...
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2 votes
0 answers
84 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 ...
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4 votes
1 answer
133 views

Proof for rotation number $\operatorname{rot}(K) = \sum \limits_C \omega_C(K) - \sum \limits_p \operatorname{ind}_p(K)$?

I need the following statement for a proof I am working on. It seems so simple and I'd rather have it ready to be cited instead of spending a page proving it (I found one for this statement), but ...
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3 votes
0 answers
188 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 ...
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4 votes
0 answers
50 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 ...
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2 votes
0 answers
156 views

Does the intersection of two moving curves contain a continuous curve?

Suppose we have two plane curves, $A$ and $B$. They are continuous, but they need not be smooth or "nice" in any other way. The lines cross each other once. Now $A$ undergoes a continuous ...
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1 vote
0 answers
55 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 ...
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14 votes
2 answers
843 views

Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

I've posted this question on Math Stack Exchange, but I want to bring it here too, because 1) the proof seems missing in the literature, although they are some sporadic mentions and 2) maybe it ...
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2 votes
1 answer
82 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 ...
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6 votes
2 answers
339 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 ...
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2 votes
0 answers
97 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 $r$...
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1 vote
0 answers
61 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(...
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1 vote
0 answers
86 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, $t$, $n$ ...
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4 votes
1 answer
321 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\...
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2 votes
1 answer
165 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 ...
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11 votes
2 answers
245 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^...
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5 votes
0 answers
309 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 ...
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0 votes
0 answers
54 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 $\Omega$...
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6 votes
2 answers
345 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 ...
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77 votes
4 answers
3k views

Does this property characterize straight lines in the plane?

Take a plane curve $\gamma$ and a disk 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 $\...
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2 votes
0 answers
44 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 ...
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22 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 ...
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1 vote
1 answer
282 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 ...
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25 votes
5 answers
2k 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 ...
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  • 395
4 votes
0 answers
83 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 ...
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19 votes
0 answers
790 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 ...
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-1 votes
1 answer
213 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 ...
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2 votes
1 answer
341 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)...
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1 vote
0 answers
199 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 ...
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3 votes
1 answer
100 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 ...
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3 votes
1 answer
80 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 ...
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  • 273
3 votes
1 answer
157 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 ...
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0 votes
0 answers
103 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 $X \times X$ ...
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2 votes
1 answer
163 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) ...
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5 votes
0 answers
97 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 ...
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  • 17k
1 vote
2 answers
118 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}{...
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1 vote
1 answer
299 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 ...
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2 votes
1 answer
110 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 \...
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4 votes
1 answer
148 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 $\textit{F}$ ...
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  • 395
9 votes
0 answers
208 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{...
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3 votes
0 answers
73 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 $\...
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1 vote
0 answers
170 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/...
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7 votes
1 answer
564 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 ...
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5 votes
0 answers
205 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 ...
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  • 395
4 votes
1 answer
193 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?
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3 votes
1 answer
217 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 $\...
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8 votes
1 answer
371 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 ...
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6 votes
1 answer
461 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 ...
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