# 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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### 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 ...
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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### 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 ...
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 ...
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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### $(n-2)$-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 ...
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 ...
1 vote
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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 ...
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 ...
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 ...
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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### 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
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### 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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1 vote
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### 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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### 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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### 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 ...
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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### 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 \$\...