Questions tagged [curves-and-surfaces]
A surface is a generalization of a plane which needs not be flat, that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line
373 questions
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Variety of conics touching a given plane sextic
To continue with the topic of Plane sextics admitting a conic with 6 double-contact points, I would like to understand the variety of conics touching a given (irreducible) projective plane sextic ...
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Zindler curves and floating bodies in the plane
Given a planar body K bounded by a simple continuous curve C that is divided into two parts of equal area by a chord of C included in the interior of K in any direction, and that all these "...
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Representation of surface group where simple curves are mapped to non-diagonalizable elements
Let $S$ be a closed orientable surface of genus at least $2$. Does there exist $n\geq3$ and a representation $\rho:\pi_1(S)\to\mathrm{GL}_n(\mathbb{C})$ such that for every simple closed curve $\gamma\...
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Second fundamental form and geodesic curvatures of frame lines
Consider a smooth surface $S$ in $\mathbb{R}^3$, if necessary simply connected, parametrized by $\mathbf{r}(u,v)$. With the second fundamental form written as $\text{II}=L\:\mathrm{d}u^2+2M\:\mathrm{d}...
- 174
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Amsler surface to a pseudospherical surface of revolution
Consider an Amsler surface—that is, a pseudospherical surface containing two straight lines that intersect at a point on the surface. It is known that such a surface is uniquely determined by the two ...
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Software that can squeeze a surface
Let $\Sigma$ be an embedded surface in $\mathbb{R}^3$ subject to certain constraints (for example, its normal curvatures lie between $\pm 1$).
Now imagine that the space $\mathbb{R}^3$ is filled with ...
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What curve maximizes the Levy area?
The Levy area of a $C^1$ curve $f:[0,\infty)\to \mathbb R^2$ is defined to be $$L_f(t):=\int_0^t (f_1(s)f_2'(s)-f_2(s)f_1'(s))ds. $$ It is called Levy area because by Green's theorem, it is twice the ...
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Smallest curvature needed to connect two points with given tangent
I have two points and their corresponding tangents in the 3D space. Is there a way to compute the minimum curvature needed to connect these two points smoothly? The line between these two points ...
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Time-smoothing of an isotopy
Let's consider the following problem: we have two smooth manifolds, $N$ and $M$, and let's suppose we have a continuous embedding $h \colon N \hookrightarrow M$. Under some conditions (for example if $...
- 147
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Space filling curve with some extra properties
Is there a sequence of simple closed curves $\gamma_n: [0,1] \to [0,1]^2$, $\gamma_n(0)=\gamma_n(1)$, which uniformly converges to a surjective continuous function $\gamma_{\infty}:[0,1]\...
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The nature of the fixed component implies the divisor is not reduced
I'm reading this article https://arxiv.org/pdf/math/0302045 . Page 20, they claim :
"Suppose
that $e \ge 3$. Then, since $D_{2}$ is linearly equivalent to $4C_{0} + (2e + 2)f$, $D2$ has $2C_{0}$ ...
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Area inside curve shortening flow [closed]
Show that the area of the inside region of a simple closed curve in the plane along
the curve shortening flow is given by
\begin{equation*}
A(\tau) = -2\pi\tau + A(\tau = 0)
\end{equation*}
As a ...
- 111
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Rolling characterization of the sphere
I have been reading A remark concerning a mechanical characterization of the sphere, where the author proves the following result:
Theorem. A compact oriented surface of $\mathbb{R}^3$ is a sphere if ...
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Entropy and stretch factor of pseudo-anosov diffeomorphism (reference request)
Let $f$ be a pseudo-anosov diffeomorphism of an $n$-punctured sphere, with $n\geq 4$.
I think it should be true that $h_{top}(f) = \ln \lambda_f$, where $\lambda_f$ is the stretch factor of $f$. ...
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Converse of Scherk–Segre theorem on the number of vertices of a convex space curve
It is well known that any smooth simple closed convex curve $\gamma$ in $\mathbb{R}^{3}$ that meets no plane in more than 4 points has exactly 4 vertices, i.e., points of vanishing torsion; here "...
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