Questions tagged [curves]
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122 questions
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What shapes can roll down tailor-made curves, without jumping off or behaving chaotically?
Background
It is well-known that the disk can roll down various curves, including the straight line and the Brachistochrone curve. The latter is the curve along which both a point rolls down the ...
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Sum of two triangles in a projective plane modulo a conic
Given a conic $C$ in the complex projective plane, say $C=\{c:=x^2+y^2+z^2=0\}$, and two “triangles” (given as zeros of products of 3 linear forms $\ell=\{ax+by+cz\}$) $\ell_1\ell_2\ell_3$, $\ell'_1\...
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Divide angles by coefficients relate to Fibonacci sequence
In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle $(...
- 1,776
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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 $\...
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3
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Difference in length of two dimensional concentric closed paths
Two bicyclists ride side by side around a smooth loop with the outside bicyclist a distance of $D$ from the inside bicyclist.
How much further does the outside bicyclist ride?
If the loop is a circle, ...
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2
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Is a simple closed curve always a free boundary arc?
Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points?
For a simple closed curve $\...
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3
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Inflection point calculation for cubic Bézier curve encounters division by zero
I've been working on finding the inflection points of a cubic Bezier curve using the method described in a paper Hain, Venkat, Racherla, and Langan - Fast, Precise Flattening of Cubic Bézier Segment ...
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80
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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)...
- 255
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113
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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 &...
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Are any embeddings $[0,1]\to\mathbb{R}^3$ topologically equivalent?
Suppose we are given embeddings $f_1,f_2:[0,1]\to\mathbb R^3$.
Does there exist a homeomorphism $g:\mathbb R^3\to\mathbb R^3$ such that $g\circ f_1=f_2$?
This question seems to be classical eighty ...
- 1,095
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105
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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 ...
- 1,097
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119
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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 ...
- 237
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Points on curves 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 a point ...
- 237
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1
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A closed curve can be homotopic to remove all intersections with a filling $\Gamma$ if it has zero geometric intersection numbers with $\Gamma$
Let $\Sigma$ be a compact oriented connected bordered surface other than the pair of pants. Let $\Gamma:=\{\gamma_i\}$ be a finite collection of simple closed curves on $\Sigma$ such that each ...
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Characterization of a non-trivial non-peripheral element of the free homotopy classes of a compact bordered surface
Let $\Sigma$ be a compact orientable connected $2$-manifold with a non-empty boundary. Let $\widehat \pi(\Sigma)$ denote the set of free homotopy classes of
curves in $\Sigma$. We say $x\in \widehat \...
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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\...
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A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 620
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66
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Representation of $x$-non-monotone curves with one intersection each by $x$-monotone curves
Take the $y$-axis and a set of $n$ curves starting from $y$-axis, labelled as $\mathcal{C}:=\{C_1,C_2,...,C_n\}$. These curves fulfill the following conditions:
The curves all have a starting point ...
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486
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There exists differentiable curves arbitrarily close to the continuous ones
Let $M$ be a Riemannian manifold; if $d$ is the distance on $M$, we can consider the distance $D$ between any two continuous curves given by $D(c, \gamma) = \max _{t \in [0,1]} d(c(t), \gamma(t))$.
...
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Rational even polynomials maximally tangent to the unit circle
This question is motivated by College Mathematics Journal problem 1196, proposed by Ferenc Beleznay and Daniel Hwang. My solution to this problem (pre-publication version here) uses Chebyshev ...
- 11.2k
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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 ...
- 13.2k
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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 ...
- 1,241
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129
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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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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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303
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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 ...
- 1,168
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$(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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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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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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1
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209
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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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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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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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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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111
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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, $t$, $n$ ...
- 11
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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\...
user124234
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1
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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 ...
user61586
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2
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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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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 ...
- 1,084
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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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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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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 $\...
- 4,459
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50
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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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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 ...
- 15.8k
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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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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 ...
- 405
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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 ...
- 4,862
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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 ...
- 5,215
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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 ...
- 5,998
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1
answer
385
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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)...
- 5,998
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vote
0
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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 ...
- 5,998
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1
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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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