For questions about surfaces.

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38
votes
4answers
1k views

How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A ...
13
votes
3answers
2k views

Cutting a Möbius strip down the middle

Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right ...
7
votes
1answer
110 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
4
votes
1answer
108 views

How to find cubic non-snarks where the $\min(f_k)>6$ on surfaces with $\chi<0$?

Henning once told me that, [i]t follows from the Euler characteristic of the plane that the average face degree of a 3-regular planar graph with $F$ faces is $6-12/F$, which means that every ...
1
vote
2answers
333 views

Identify and sketch the quadric surface?

I'm stuck trying to figure out which type of quadric surface this equation is: $$\dfrac{x^2}{16} - \dfrac{y^2}{9} - \dfrac{z^2}{1} = 1$$ I have narrowed it down to a hyperboloid, but cannot ...
0
votes
0answers
151 views

Optimal number of points for integration over the surface of a sphere

My objetive is to calculate overlapping between spheres. What is the minimum number of points to allocate in the surface of a sphere, so we can perform numerical integration over each sphere with the ...
17
votes
2answers
359 views

How to determine that a surface is symmetric

Given a surface $f(x,y,z)=0$, how could you determine that it's symmetric about some plane, and, if so, how would you find this plane. The special case where $f$ is a polynomial is of some interest. ...
20
votes
3answers
246 views

Is an isometric embedding of a disk determined by the boundary?

Suppose we cut a disk out of a flat piece of paper and then manipulate it in three dimensions (folding, bending, etc.) Can we determine where the paper is from the position of the boundary circle? ...
1
vote
0answers
101 views

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the ...
7
votes
4answers
1k views

Drawing a thickened Möbius strip in Mathematica

I would like to have Mathematica plot a "thickened Möbius strip", i.e. a torus with square cross section that is given a one-half twist. Ideally, I would like this thickened Möbius strip to be ...
6
votes
2answers
46 views

1-dimensional foliation on a surface

Is it possible to find a 1-dimensional nonsingular foliation on an orientable surface with one boundary component such that lines of the foliation are transverse to the boundary?
5
votes
1answer
201 views

Limit $\lim_{x\rightarrow x_0, x\in M} \int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac{-1}{||y-x||} dS_y$

Ok I had a question I think I can almost answer it but I miss one step: Let $\partial M$ be a closed surface in $\mathbb{R}^3$, $x_0 \in \partial M$ than show this limit: ...
4
votes
1answer
309 views

Covariant derivative on hypersurface in $\mathbb{R}^n$

I saw in a talk that a surface gradient of $f:M \to \mathbb{R}$ where $M$ is a hypersurface in $\mathbb{R}^n$ defined as $$\nabla_M f = \nabla f - (\nabla f \cdot N)N$$ where $N$ is the unit normal ...
2
votes
2answers
138 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
2
votes
1answer
139 views

Does the uniqueness of solutions to convex optimization with linear constraints hold in n>3 dimensions?

This is a repost of an earlier question, where I think I was not clear enough in what I was asking: I am examining the following optimization problem, for which I would like to know if, when a ...
6
votes
2answers
339 views

isolated non-normal surface singularity

I am looking for an isolated non-normal singularity on an algebraic surface. One obvious example occurs to me: the union of two $2$-dimensional affine subspaces of $\mathbb{A}^4$ which meet in a ...
5
votes
3answers
233 views

Why is $\pi r^2$ the surface of a circle

Why is $\pi r^2$ the surface of a circle? I have learned this formula ages ago and I'm just using it like most people do, but I don't think I truly understand how circles work until I understand why ...
5
votes
1answer
99 views

Is this function bounded? Next question about integral $\int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac1{||y-x||} dS_y$.

Let $\partial M$ be $C^2$ closed surface in $\mathbb{R}^3$, $M$ is open. Show that $$ f(x) = \frac{\int_{\partial M} \left| \frac{1}{||y-x||} n_y \cdot \nabla_y \frac{1}{||y-x||} \right| ...
3
votes
0answers
35 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
3
votes
2answers
121 views

What's the K-group of a surface?

What's the K-group of a surface? I also want to know how to calculate such group and if there is a explicit characterization of the generators.
3
votes
2answers
133 views

Equivalence of two definitions of differentiablitity on Regular Surfaces

When dealing with differentiable surfaces one defines a function $f:S\rightarrow \mathbb{R}$ as being differentiable if its expression in local coordinates is differentiable. But one could also define ...
3
votes
4answers
377 views

parametrization of surface element in surface integrals

I don't understand this How $ dS = \sqrt{ \left ( \partial g \over \partial x\right )^2 + \left ( \partial g \over \partial y\right )^2 + 1 } \; dA \; \; $ ?? Is $ dA = dx\times dy$??
2
votes
1answer
96 views

Topological surface thought experiment

Imagine a two-dimensional version of you lives on some compact, connected surface (orientable or non-orientable). How would you figure out on which surface you are living? Are there experiments you ...
2
votes
1answer
114 views

Is conformal equivalence the same as topological equivalence?

Is it true that if I take two surfaces that are topologically equivalent, I can find a conformal mapping between them?
2
votes
1answer
793 views

The Gaussian and Mean Curvatures of a Parallel Surface

This is a homework problem from do Carmo. Given a regular parametrized surface $X(u,v)$ we define the parallel surface $Y(u,v)$ by $$Y(u,v)=X(u,v) + aN(u,v)$$ where $N(u,v)$ is the unit normal on $X$ ...
2
votes
1answer
104 views

Is the intersection of the diagonal with a graph always transverse in characteristic zero

Let X be a projective smooth connected curve over $\mathbf{C}$. Let $f:X\to X$ be a non-constant morphism. Is the intersection of the diagonal $\Delta_X$ and the graph $\Gamma_f$ on $X\times X$ ...
1
vote
1answer
200 views

Pinched torus generalization

The pinched torus is homeomorphic to a sphere with two (different) points identified.           What is the name and topological structure of the ...
0
votes
1answer
230 views

compute principal directions of a cylinder

I calculated the parametric equation of a cylinder, $$x(u,v)=a\cos(u)$$ $$y(u,v)=a\sin(u)$$ $$z(u,v)=v$$ I do not know how to calculate principal directions ? I am not sure what it means neither ...