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

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Cross product of two infinitesimal bendings

Let $M$ be a smooth (embedded or immersed) surface in $\mathbb{R}^3$. Let $Z_1,Z_2$ be two vector fields along $M$, thought of as $\mathbb{R}^3$-valued functions, satisfying the following differential ...
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Singularities of arithmetic surface

I have a problem understanding the discussion of example 8.3.54 in Liu's Algebraic Geometry and Arithmetic Curves. The setting is the following: We have a DVR with uniformiser $t$, characteristic of ...
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Can every smooth space curve be realized as an origami curved crease?

Many years ago, Ron Resch told me that he proved that every smooth simple space curve $C$ could be realized as a curved crease $\gamma$ in the interior of a piece of paper. He never published this (as ...
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3 votes
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Smoothness of ruled surface (asymptotic) parameterisations

A ruled surface $S$ shall be defined as surface consisting of straight line segments. It is commonly known (cf. [BER, p.362] or [STR, p.93] - bibliography at the end) that a ruled surface allows for a ...
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2 votes
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Why is $H$ the standard notation for mean curvature?

I am curious about the origin of the notation $H$ to denote the mean curvature of a surface in $\mathbb{R}^{3}$. I suppose that the symbol $K$, which is commonly used to denote the Gaussian curvature, ...
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Smoothness of the asymptotic parametrization of a ruled surface

Let $S$ be a smooth developable surface in $\mathbb{R}^{3}$. It is well known that, if $S$ is free of planar points, then it admits a local parametrization of the form $$\begin{align} \sigma \colon I \...
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rational curves over K3 surfaces over $\mathbb{Q}$

There are many partial results towards the following conjecture: Every projective K3 surface over an algebraically closed field contains infinitely many integral rational curves. My question is: is ...
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Where's the negative section of a deformation of a Hirzebruch surface?

As in Deformations of Hirzebruch surfaces and toric action, the Hirzebruch surface $F_n$ can be deformed into $F_{n-2m}$ ($0<2m\leq n$) under the fibration given by $$ M=\{([x_0:x_1],[y_0:y_1:y_2],...
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Essential simple closed curves in a torus [closed]

Definition: By a closed curve in a surface $S$ we will mean a continuous map $S^1 \to S$. We will usually identify a closed curve with its image in $S$. A closed curve is called essential if it is not ...
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About isotopy of simple close curve

In the Primer mapping class group by farb Margalit. We have : Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\alpha$ is isotopic to $\beta$ if ...
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The classification of developable surfaces: Are these statements equivalent?

This is a cross-post from MSE (https://math.stackexchange.com/q/4330772/242708). I thought to know very well the answer to the classification problem for developable surfaces, so I sought for some ...
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An equivalent characterization of conical surfaces

Consider an $n-1$ dimensional surface in $\mathbb{R}^n$. If the tangent plane at any point of this surface always passes through the origin, can we show that the surface must be a conical surface? I ...
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Is the Moebius strip Riemannian homogeneous?

Let $ M $ be the Moebius band. In other words, the total space of the nontrivial line bundle over the circle. Can we equip $ M $ with a metric such the the isometry group acts transitively? My ...
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3 votes
1 answer
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Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$?

Let $\gamma(s)$ be a unit speed planar curve with $\kappa(s)$ as its curvature. Now what is $\int \frac{1}{\kappa}ds$ and geometrically which things it represents?
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What is the status of Jordan's theorem in constructive mathematics in the language of locales?

By constructive mathematics in this matter we mean intuitionistic ZF (*). In the language of locales, the Jordan curve can be defined as $f\colon S^1 \to X$ such that "if $U \cap V = \varnothing$,...
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Separability of graph component embeddings

I have an undirected graph. I also have an embedding of the graph in $\mathbb{R}^3$. Assume that the graph has 2 connected components. I want to know whether there exists a plane (or better - any ...
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Can we define surface integral on 'bad surface'?

We can define a surface integral on a piecewise smooth surface, but if the surface is not piecewise smooth can we use measure theory to generalize the definition of surface integral? And does Stokes ...
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How can I find the area of a rectangle created by the spiral r=theta at a certain theta? [closed]

I would like to have a function that gives the area of a rectangle at a certain theta of the spiral r=theta. The height of the rectangle is the y value of the point on the spiral and the base of the ...
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2 votes
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understanding the definition of subgroup of the Grothendieck-Teichmuller group

Definition. Let $\widehat{G T}^{1}$ be the set of elements $f$ in the derived subgroup $\hat{F}_{2}^{\prime}$ of $\hat{F}_{2}$ such that $x \mapsto x$ and $y \mapsto f^{-1} y f$ extends to an ...
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Pushing figures into holes

Let $\gamma_1,\gamma_2:[0,1]\to \mathbb{R}^2$ - smooth curve, $\gamma_i(0)=\gamma_i(1)$, $X_1$ and $X_2$ are the areas bounded by the corresponding curves. . Suppose we have an $X_1 $-shaped hole, and ...
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What's the name of this surface: $z = \exp(xy)$ [exponentialoid?] [closed]

When studying the real value exponential, I encounter the surface $z = e^{x\cdot y}$ but I don't know if it has a name. I've created a 3D applet to explore it. When I cut it by the plane $$ (x-x_0)\...
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2 votes
1 answer
224 views

Continuity of the perimeter of level sets w.r.t. level function

Working with the level set method introduced by Osher & Sethian in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the ...
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Continuity of Hausdorff measure on level sets

Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that: $\bullet$ $\phi^{-1}(0)\neq\emptyset$; $\bullet$ $\nabla\phi(x)\...
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4 votes
1 answer
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Classes of curves closed under Minkowsky sum

Let $C$ be a class of plane curves, regarded as subsets of $\mathbb{R}^2$ (parametrization won't matter), I'm thinking for example of splines or algebraic subsets. Let $D$ be a class of topological ...
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2 votes
1 answer
199 views

When does the Hirzebruch surface have a nef anticanonical divisor?

Let $\mathcal H_r=\mathbb P (\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(r))$ be a Hirzebruch surface for some $r\in\mathbb Z$. As a toric variety, the fan structure is spanned by $(-1,0)$,...
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6 votes
2 answers
231 views

Necessary and sufficient curvature condition for a regular planar curve to be simple and closed

Given a smooth, $2\pi$-periodic function $\kappa(s)$, the associated planar curve $\gamma(s)$ for which $\kappa(s)$ is the (signed) curvature, is uniquely determined up to Euclidean invariance: a ...
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5 votes
1 answer
171 views

When does a spherical curve equal its tangent indicatrix?

Given a smooth regular curve $\gamma$ in $\mathbb{R}^{3}$, one defines the tangent indicatrix of $\gamma$ to be the spherical curve $\gamma'/\lVert \gamma'\rVert$. It is then natural to look for ...
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3 votes
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Existence of developable ribbonization of a surface

Let $S$ be a smooth compact surface embedded in $\mathbb{R}^{3}$. It is well-known that there exists a triangulation of $S$. I am considering an alternative way of approximating $S$, where instead of ...
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4 votes
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When is the cut locus a tree?

Let $\gamma: \mathbf{S}^1 \to \mathbf{R}^2$ be a simple, closed curve. Assume additionally that $\gamma$ is regular, of class $C^2$ at least. The complement of $\gamma$ in $\mathbf{R}^2$ has two ...
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Understanding sheaves on normalisation of a curve: $v_* \mathcal{O}_{\tilde{C}} / \mathcal{O}_C$

Let $(C, \mathcal{O}_C)$ be a reduced irreducible curve and $(\tilde{C},\mathcal{O}_{\tilde{C}})$ its normalisation with $v : \tilde{C} \rightarrow C$. Then we have an imoprtant skyscraper sheaf $v_* \...
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0 votes
1 answer
127 views

"Arc" length parametrization for surfaces

If we have a function $\phi:\Omega\subseteq\mathbb{R}^2\to\mathbb{R}$, $\phi\in C^2(\Omega)$ is there a way to find a function $d:\Omega\to\mathbb{R}, d\in C^2(\Omega)$ so that: $|\nabla d(x,y)|=1,\ \...
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2 votes
1 answer
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Hyperbolic length of curve that does not enter a collar

Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that ...
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9 votes
3 answers
644 views

Is there the longest geodesic?

Given a closed 2-surface $M$ together with a Riemannian metric $g$. We pick a free homotopy class $\gamma \in \pi_1(M)$ and consider the set $C(\gamma)$ of all closed geodesics homotopic to $\gamma$. ...
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2 votes
0 answers
133 views

Degree $4$ curves on K3 double covers of Del-Pezzo surfaces

Let $S$ be a smooth del-Pezzo surface and $\pi : X \longrightarrow S$ be the double cover of $S$ ramified in a smooth section of $-2K_S$. Going through the classification of del-Pezzo surfaces, one ...
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13 votes
1 answer
306 views

Are any of these complex surfaces ever projective?

Let $C$ and $T$ be compact connected Riemann surfaces (or: smooth projective connected curves over $\mathbb{C}$) of genus at least two and let $X:=C\times T$. Let $(c,t)$ be a point of $X$, and let $...
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5 votes
1 answer
171 views

Representing relative homology classes orientable surfaces with boundary

Let $S$ be compact oriented surface without boundary. Then it is a classical result that a primitive class $\gamma \in H_1(S; \mathbb{Z})$ is always represented by a simple closed curve. It implies ...
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2 votes
0 answers
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On cycloids and other roulettes

It is well known that the cycloid is the curve traced by a point on a circle as the circle rolls along a line without slipping. Consider wheels with smooth convex shapes (not necessarily circular) ...
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2 votes
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intuition behind shape optimization using Hadamard's method

I'm trying to understand the intuition behind shape optimization using Hadamard's method. Please consider the following simple example: Let $\lambda$ denote the Lebesgue measure on $\mathcal B(\...
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3 votes
0 answers
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Shape derivative of boundary integrals and differentiability of the integrand on a tubular neighborhood

Let $d\in\mathbb N$, $U\subseteq\mathbb R^d$ be open, $$\mathcal A:=\{\Omega\subseteq\mathbb R^d:\Omega\text{ is bounded and open},\overline\Omega\subseteq U\text{ and }\partial\Omega\text{ is of ...
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6 votes
2 answers
256 views

An abstract characterization of line integrals

Let $M$ be a smooth manifold (endowed with a Riemann structure, if useful). If $\omega \in \Omega^1 (M)$ is a smooth $1$-form and $c : [0,1] \to M$ is a smooth curve, one defines the line integral of $...
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3 votes
0 answers
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The space of rearrangements of a plane curve

I learned of the paper "Closing curves by rearranging arcs" by L. Alese (arXiv link) by a post on Reddit and was curious about related questions and generalizations. Here a rearrangement of ...
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1 vote
2 answers
203 views

When is the inside of a Jordan curve open? [closed]

I'm working purely on intuition here. The Jordan curve theorem states that a Jordan curve separates the plane into a bounded component and an infinite component. For toy curves, it seems like this ...
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25 votes
5 answers
2k views

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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1 vote
1 answer
203 views

Non-isotrival fiber bundle over compact Riemann surface

In this paper, Kodaira constructed a fiber bundle $\Phi:M_{m,n}\to S$ from a compact complex surface $M_{m,n}$ to a compact Rieman surface $S$ of genus $>0$. In particular, (on p.212) for any point ...
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1 vote
0 answers
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Difference of two functions with constant mean curvature

Define the set $\Omega := (-\epsilon,\epsilon) \times (-1,1)^{n-1}$, and define $\Gamma := \{-\epsilon,\epsilon\} \times (-1,1)^{n-1} \subset \partial \Omega$. Suppose I have two functions $u,v \in C^...
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2 votes
1 answer
403 views

Geometric/Algebraic intersection numbers of curves on surfaces

I have the following problem, and struggling to find some references. Suppose I start with a homology class of a curve on a closed genus $g$ surface $$h=(a_{1},b_{1},\dots,a_{g},b_{g})\in H_{1}(\...
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2 votes
1 answer
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Coordinates for Laminations: geometric versus shear

Let $S$ be an orientable surface with a triangulation T. A lamination $\ell$ is a simple closed curve on $S$, up to isotopy. We will assume that $\ell$ is drawn in such a way that it intersects the ...
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3 votes
0 answers
114 views

Chord of fixed length traveling around a Jordan curve

Let $C$ be a Jordan curve with nice enough properties whenever necessary (e.g. smooth, or just rectifiable, perhaps). I am interested in knowing how long can a chord be that "traverses" the ...
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19 votes
0 answers
790 views

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 ...
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  • 3,594
6 votes
1 answer
156 views

Stationary phase in spherical integral

I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that: If $\lambda\gg 1$...
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