For questions about surfaces.

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2
votes
1answer
74 views

Creating an ellipsoidal 3D surface

I am trying to find the equation of a 3D ellipsoidal surface. I have thought of two approaches which are schematically shown below: By revolving an elliptical arc over a 3D elliptical path: Or by ...
7
votes
4answers
295 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 ...
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 ...
15
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
112 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
110 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
483 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
155 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
386 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
265 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
107 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 ...
0
votes
1answer
41 views

The equation of a 3D surface bounded by 3 known elliptical curves

I am trying to find the equation of a 3D surface as illustrated below. The boundaries of this surface is comprised of two planar elliptical arcs $AB$ and $AC$ as well as a 3D arc $BC$ which is a 3D ...
0
votes
0answers
24 views

Non-standard 3D rotation of a set of points [duplicate]

I want to create a 3D surface as shown in the figure below. Toward this, I thought if I rotate a set of points in $xy$-plane on a elliptical arc I may be able to get such a surface. I was thinking of ...
6
votes
2answers
67 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
206 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: ...
5
votes
2answers
371 views

What function has a graph that looks like this?

I delete my file which I used to produce this graph. Does anybody have some idea how to produce it again? Thanks for a while.
4
votes
2answers
82 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
4
votes
1answer
347 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 ...
4
votes
2answers
414 views

Surface area of a flexible tube

Consider the hollow tube formed by sweeping a circle of radius $r(t)$ along a curve $\gamma(t)$ in $\mathbb{R}^3$; in other words, the set of points $$S=\{\gamma(t) + r(t) \hat{n}\quad \vert\quad ...
3
votes
1answer
185 views

A 3-manifold with fundamental group isomorphic to a surface group.

Let $M$ be a 3-manifold (the case I am interested is $M$ closed orientable connected hyperbolic); suppose $\pi_1 (M)$ is isomorphic to the fundamental group of a (closed orientable connected) surface ...
2
votes
4answers
211 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
176 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 ...
7
votes
2answers
383 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
1answer
119 views

Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
5
votes
1answer
102 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| ...
4
votes
0answers
99 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
0answers
125 views

Squeezed cylinder parametrization

A Cylinder is such a common surface. But is there a parametrization for an isometrically $ R^2 $ bent cylinder whose major and minor dimensions are along x, y axes? I used an approximation to ...
3
votes
1answer
1k 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 $X$ ...
3
votes
2answers
125 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
141 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
428 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
105 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
126 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
146 views

About fibers of an elliptic fibration.

Consider the following pencil of cubics: $\lambda C_1+ \mu C_2$ where $C_1=y^2z$ and $C_2=x(x^2+2xz+z^2)$ and the elliptic fibration $\tilde X \rightarrow \mathbb P^1$ induced by the blow-up of ...
2
votes
1answer
111 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
2answers
68 views

Describe a twisted parabolic trough

I want to describe a parabolic trough of the form $z=x^2$ and give it a twist, like a torsion in $y$ direction. Does anybody know how I can do that? Imagine this is the trough and the $z$ direction ...
1
vote
1answer
68 views

Show that $(\textbf{S}^*\textbf{B})(u,v)=\textbf{B}(\textbf{S}(u,v))\cdot \textbf{N}(u,v) \ du \wedge dv$

Let $\textbf{S}(u,v):[0,1]^2 \rightarrow \mathbb{R}^3$ be a singular $2$-cube which is smooth. Note that $0 \leq u,v \leq 1$. Let $B(\textbf{r})=B_x \ dy \wedge dz + B_y \ dz \wedge dx + B_z \ ...
1
vote
1answer
102 views

Nonorientable surfaces: genus or demigenus?

The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, ...
1
vote
0answers
252 views

$N$ equally spaced points on an ellipsoid

I would like to find a algorithm for determining the $(x,y,z)$ co-ordinates for evenly distributed $N$ points on the surface of an ellipsoid. These points must be spaced from its nearest neighbour ...
1
vote
1answer
1k views

Finding surface area of cone inside a cylinder

So I am presented with the following problem: Find the surface area of the cone $z=\sqrt{ x^2 + y^2} $ that lines inside the cylinder $x^2 + y^2 = 2x$. Im pretty sure a double integral is involved, ...
1
vote
1answer
229 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
63 views

Gauß and mean curvature

I was wondering whether the Gauß, mean curvature and shape operator of a surface actually depend on the chosen parametrization? Under a reparametrization of $f: \Omega \subset \mathbb{R}^2 ...
0
votes
1answer
41 views

Elongation of edges of Moebius strip

Find length of edges of Moebius strip formed by cutting along length and re-joining from a circular cylinder segment: $( a \cos \theta, a \sin \theta, z), ( \theta, 0, 2 \pi ),( z,0,b) $ after half ...
0
votes
0answers
47 views

Lattice representation of the Klein bottle

I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems. To ...
0
votes
0answers
29 views

sphere parametrization

We have the standard spherical surface parametrization in which one set describes geodesics (longitudes) and another ( latitudes/parallels). What parametrization may be possible for both sets to be ...
0
votes
1answer
301 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 ...