Questions tagged [curves]

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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 ...
Malkoun's user avatar
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15 votes
0 answers
310 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
9 votes
0 answers
238 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
5 votes
0 answers
370 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 ...
Leo Herr's user avatar
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5 votes
0 answers
100 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
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5 votes
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225 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
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5 votes
0 answers
329 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
5 votes
0 answers
324 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
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5 votes
0 answers
409 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
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4 votes
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63 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 ...
Vu Thanh Tung's user avatar
4 votes
0 answers
97 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 ...
Ivan Meir's user avatar
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4 votes
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112 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 ...
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4 votes
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142 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
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3 votes
0 answers
256 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 ...
XT Chen's user avatar
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3 votes
0 answers
151 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 $r$...
Babai's user avatar
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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
3 votes
0 answers
113 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
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2 votes
0 answers
99 views

Vanishing of Goldman bracket requires simple-closed representative?

Let $\Sigma$ be a connected oriented surface, and $[-,-]\colon \Bbb Z\big[\widehat\pi(\Sigma)\big]\times Z\big[\widehat\pi(\Sigma)\big]\to Z\big[\widehat\pi(\Sigma)\big]$ be the Goldman Bracket. Note ...
Random's user avatar
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2 votes
0 answers
107 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 ...
TribalChief's user avatar
2 votes
0 answers
50 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 ...
Klaus Weihrauch's user avatar
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
  • 12.6k
1 vote
0 answers
93 views

Cokernel of map of dual of sheaves of differentials/deformations

Let $C$ be a nodal projective curve over an algebraically closed field of genus at least $2$. There are two natural "differential objects" one can consider: The sheaf of differentials $\...
Matthias's user avatar
  • 183
1 vote
0 answers
67 views

Biot-Savart-like integral for a toroidal helix

The following problem originates from Physics, so I apologize if I will not use a rigorous mathematical jargon. Let us consider a toroidal helix parametrized as follows: $$ x=(R+r\cos(n\phi))\cos(\phi)...
AndreaPaco's user avatar
1 vote
0 answers
97 views

Curvature of randomly generated B-spline curve

I am working on Bayesian statistical estimation of parameters (control points) of closed B-spline curve bounding an object on a an image. The problem is that I require those curves to not be much &...
MatEZ's user avatar
  • 31
1 vote
0 answers
119 views

More on points on a curve of genus 3

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find two ...
user95246's user avatar
  • 237
1 vote
0 answers
40 views

Fibrewise coordinates in a neighborhood of a graph of a continuous curve

Let $M$ be a smooth manifold, $\dim M=n$ and $\gamma:[0;1]\to M$ be continuous. Is it true that there exists local coordinates $(y^0,\ldots,y^n)$ in a neighborhood $V$ of the graph $\{(t,\gamma(t)),t\...
Lev's user avatar
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1 vote
0 answers
34 views

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 ...
Manfred Weis's user avatar
  • 12.6k
1 vote
0 answers
87 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 ...
Nandakumar R's user avatar
  • 5,473
1 vote
0 answers
71 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(...
Jacob Schwartz's user avatar
1 vote
0 answers
103 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, $t$, $n$ ...
Iruka's user avatar
  • 11
1 vote
0 answers
229 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 ...
user267839's user avatar
  • 5,948
1 vote
0 answers
196 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
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
  • 5,019
1 vote
0 answers
58 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
1 vote
0 answers
509 views

support of embedded points in a curve

Let $C \subset \mathbb{P}^n$ be an one dimensional scheme. Suppose that $C$ decomposes as the union of a Cohen Macaulay reduced curve $\tilde{C}$ (in particular $\tilde{C}$ does not have embedded ...
NotNow's user avatar
  • 103
0 votes
0 answers
65 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 $\Omega$...
Math-Physics's user avatar
0 votes
0 answers
113 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
0 votes
0 answers
83 views

Dimension of the Representation of the Suzuki and Ree Groups?

What are the dimension of the group representation of $^2B_2$ and $^2G_2$? All what I know is that the first is 4 and the second group has two representation of dimension 7 and 13. Are there any?
abd's user avatar
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