Questions tagged [curves]
The curves tag has no usage guidance.
122 questions
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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 $\...
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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 ...
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2
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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. ...
- 2,558
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3
answers
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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
19
votes
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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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2
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Is this statement which relates the Fourier transform of a function to its singularities correct?
I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
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4
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Moduli space of genus 2 curves
Does any body know any reference in which the geometry of compactified moduli space of genus two curves ( Which is a three dimensional variety/stack/...) has been studied?
16
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1
answer
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Can a shape rolling inside itself reproduce that shape?
Q. Is the circle the only shape that, when rolling inside itself,
has a point that draws out a scaled copy of itself?
Let $C$ be a simple, closed, smooth curve in the plane.
(Likely "smooth" can be ...
- 151k
15
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1
answer
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Number of curves over a finite field
Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...
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0
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330
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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 ...
- 18.6k
14
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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 ...
13
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Closest point on Bézier spline
Given a two-dimensional cubic Bézier spline defined by 4 control-points as described in the Wikipedia entry, is there a way to solve analytically for the parameter along the curve (ranging from 0 to 1)...
- 243
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2
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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 ...
- 121
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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^...
- 60.5k
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3
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Automorphisms of cartesian products of curves
Let $C$ be a smooth projective curve. Is it true that
$$\textrm{Aut}(C\times C)\cong S_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$
and in case, what would be a reference for this? Thanks.
user82886
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4
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How can I find the average of two 2D curves?
I have a curve interpolation problem.
I have two closed curves that are defined on an X,Y plane. How can I define a 3rd curve that is the average of those two? Programmatically, I have a list of ...
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1
answer
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A tricky tractrix question about vertical tangents
This is raised by a recent question occurring in combinatorial geometry.
It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ (...
- 13.4k
10
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1
answer
294
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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
10
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1
answer
335
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Shimura surfaces that do not contain a Shimura curve
Let $S$ be a Shimura surface i.e. a Shimura variety with $dimS=2$. Does $S$ necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the ...
- 2,221
9
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1
answer
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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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1
answer
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A chain of six circles associated with a conic
I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems:
A chain of six circles ...
- 1,044
9
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0
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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{...
- 151
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votes
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441
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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 ...
- 45k
8
votes
2
answers
587
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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 ...
user114642
7
votes
4
answers
4k
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Generating random curves with fixed length and endpoint distance
Are algorithms already known, that generate (arbitrarily good approximations of) random curves, w.l.o.g. with unit length, and joining endpoints $(0,0)$ and $(\alpha,0)$ with $\alpha \lt1$ given?
The ...
- 13.2k
7
votes
1
answer
676
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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 ...
- 1,776
7
votes
1
answer
899
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Isotrivial families with non-zero Kodaira spencer map
Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this ...
- 171
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radius of tubular neighborhood
Hi there,
Is there any result about the calculation of radius of tubular neighborhood of submanifold inside a Riemannian manifold?
For example, given a simple smooth curve on R^2, what's the radius ...
- 583
7
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1
answer
336
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Uniformisation for non simple closed curves
Given a simple closed curve in the plane, there is a homeomorphism from the unit open disk to the interior of the curve. The homeomorphism can be taken conformal, this is the uniformisation theorem.
...
- 18.7k
6
votes
1
answer
768
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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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2
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434
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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 ...
6
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2
answers
426
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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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1
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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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6
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1
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Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)
In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid.
Could you ...
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2
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Is it possible to check two curves on birational equivalence by some computer algebra system?
I have two curves, for example hyperelliptic:
\begin{align}
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\\\
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1
\end{align}
Is it possible to check them ...
- 424
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3
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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 $p$...
- 289
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2
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783
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endomorphisms of the Jacobian of a curve
Let $C$ be a smooth, projective curve of genus >1 over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. ...
- 51
5
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0
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413
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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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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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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 ...
- 405
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333
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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 ...
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"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 ...
- 1,163
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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 ...
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544
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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
4
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2
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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 \...
- 1,097
4
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1
answer
266
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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?
- 247
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1
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578
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A conjecture like Cayley–Bacharach theorem
Let six points $A, A', B, B', C, C'$ lie on a conic and a cubic. Let a conic through $B, B', C, C'$ and meets the cubic again at $A_1, A_2$. Let a conic through $C, C', A, A'$ and meets the cubic ...
- 1,044
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1
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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 $\textit{F}$ ...
- 405
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votes
1
answer
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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 ...
- 41
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1
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A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem
I propose a conjecture variant of Cayley-Bacharach's theorem.
I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know ...
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