For questions about surfaces.

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2
votes
2answers
88 views

How to define a surface which includes three coordinate axes?

Imagine a plane with two crossing lines along $x$ and $y$ axes, and one line $y=x$. Now bend the $y=x$ line towards $z$ axis, dragging the part of that plane with it. When this line coincides with $z$ ...
0
votes
1answer
262 views

How to integrate over an arbitrarily positioned spherical cap in spherical coordinates

If you want to integrate over the SURFACE of a spherical cap that is positioned in the way it is on wikipedia, this is rather simple. since it has azimuthal symmetry you get a factor $2\pi$ and for ...
7
votes
4answers
279 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 ...
1
vote
3answers
763 views

Find internal surface area of painted cube

Suppose that a wooden cube, whose edge is $3$ inch, is painted red, then cut into $27$ pieces of $1$ inch edge. Find total surface area of unpainted? First of all, I have tried to draw the cube using ...
1
vote
2answers
144 views

Does a Möbius strip have only one shape? Or may it have different shapes?

I'm reading a book about geometry, and after thinking and viewing the Möbius strip, I want to know whether the book is right or not. The book says with a little description (that I can't write here ...
1
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0answers
245 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 ...
6
votes
1answer
81 views

isometries of the sphere

There is a theorem by Pogorelov that if a $C^2$ surface $M$ in $\mathbb{R}^3$ is isometric to the unit 2-sphere, then $M$ is itself (a rigid motion of) the sphere. What is known about isometric ...
4
votes
1answer
109 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 ...
14
votes
2answers
242 views

Area of supercircles, or how to integrate $\int_0^1 \sqrt[n]{1-x^n}dx$?

Martin Gardner, somewhere in the book Mathematical Carnival; talks about superellipses and their application in city designs and other areas. Superellipses(thanks for the link anorton) are defined by ...
0
votes
1answer
154 views

About zeros of vector fields in compact surfaces

I'm studying compact surfaces and in particular the relationship between zeros of vector fields defined on them and Euler characteristic of the surface herself. Let be $S$ a compact (smooth) surface ...
1
vote
1answer
42 views

Formula for curvature of two intersecting surfaces in terms of their normal curvature.

I have been privately reading DoCarmo recently, and have been attempting to do some of the problems. I am stuck on this one, it is problem 14 in section 3.2 for those interested. If someone could show ...
2
votes
2answers
174 views

Finding surface area - integral of $\sqrt{1+z^2}$

Sorry about this, this is more of a "am I going the right way" question, there's a surface it goes: $$x^2+y^2-z^2=1$$ Now this is nice because $x^2+y^2=r^2=1+z^2$ thus $r=\sqrt{1+z^2}$ (I want the ...
1
vote
1answer
134 views

How to define a surface $z = f(x,y)$ with flat region at centre and sigmoidally tapering towards the edges?

How do we define a continuos function $f(x,y)$ within the bounded domain $x \in [a,b]$ and $y \in [c,d]$ so that $z=f(x,y)$ has a flat surface at the centre (flat means $f(x,y)= C$, $C$ being ...
0
votes
2answers
1k views

What is a smooth surface?

What is a smooth surface in terms of tangents and normals? I read in a book that surfaces are smooth if its surface normals depend continuously on the points of that surface. I did not understand this ...
11
votes
1answer
178 views

Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic ...
1
vote
1answer
411 views

Surface area element of an ellipsoid

I would like to evaluate an integral numerically over the surface of an ellipsoid. Take an $N \times N$ grid over the parameter space $(u, v) \in [0, 2\pi) \times [0, \pi) $. A simple approximation of ...
1
vote
0answers
47 views

some question in the proof of classification of compact connected surface

Each compact connected $2$-manifold $S$ has a proper triangulation $K$, so we can order all $2$-simplices of $S$, $F_1,F_2,\ldots,F_{k-2}$ such that $F_i$ meets $F_{i-1}\cup F_{i-2} \cup \ldots \cup ...
3
votes
2answers
190 views

hypersurface evolving with tangential velocity

If a hypersurface $S_t$ evolves with velocity only in the tangential direction, is $S_t \equiv S_0$ for all $S$? This is what I have read is true (or something very similar). Can someone give me an ...
1
vote
0answers
130 views

Finding local maxima of a 2d dynamically created function

So this is a problem I am trying to solve to try and find objects in images. A given image is scanned and the coordinates of detected features are returned. With these coordinates I want to assign a ...
1
vote
1answer
57 views

Cut edge between two parametric surfaces

I want to make a model of an ultrasound field, that impinges on a test object. The shape of the sound field can be simplified as a cone and the test object is cylindric. I used the following ...
1
vote
1answer
122 views

How to use Minkowski's formula to prove Liebmann's theorem?

Let $M$ be a compact connected surface embedded in $\Bbb R^3$, and has constant Gaussian curvature. I am asked to show that $M$ is a canonical sphere by using the Minkowski's formula. Minkowski's ...
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
101 views

Computing a contraction of an exceptional divisor.

For a few days, I have been working on the following problem, from Qing Liu's book: Let $\mathcal{O_K}$ be a discrete valuation ring with uniformizing parameter t and residue characteristic $\neq ...
1
vote
1answer
260 views

Interpolating missing points in 3D data-set

Given the following x,y,z points (z is actually a signal strength indicator in dBm): ...
1
vote
1answer
98 views

Hatcher's Problem 3.2.18

How can I prove that: For the closed orientable surface $M$ of genus $g \geq 1$, show that for each nonzero $a \in H^1(M; \mathbb Z)$ there exists $b \in H^1(M; \mathbb Z)$ with $ab \neq 0$.
8
votes
1answer
120 views

Surface integral of $2x+y+2z=16$

Here's the question: Find the surface area of the part of the plane $2x+y+2z=16$ bounded by the surfaces $x=0$, $y=0$ and $x^2+y^2=64$. So, I know I have to parameterize the surface ...
1
vote
1answer
41 views

A triangular “spot function”

z = (cos πx + cos πy) represents the classical "spot function", made by square cells, used in every laser printer's halftone screening. Does anyone knows the corresponding function to produce ...
10
votes
2answers
260 views

Integral of wedge product of two one forms on a Riemann surface

I'm having trouble verifying an elementary assertion made in this answer on MathOverflow. It seems more like a math.stackexchange question, so I'm asking it here. Anyway, the assertion is as follows ...
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: ...
3
votes
1answer
36 views

Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere?

Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere? I want to know if I have any convex subset $M$ of $\mathbb{R}^n$ is it meaningful to talk about integral of second kind ...
1
vote
0answers
46 views

Calculating the area of a side of a solid (I just need a verification of the answer)

I need to calculate the area of the solid $$S=\{(x,y,z): x,y\geq 0, x^2+y^2\leq 1, 0\leq z \leq xy \}.$$ I'm not aksing you to do it for me, I'm just asking to check whether I did it right. (I would ...
6
votes
0answers
124 views

Paper cylinder inside out

My question is related with paper folding: Given a cylinder of paper it is possible to turn it inside out using folding along lines. This is a Martin Gardner recreational puzzle. Secondly, ...
2
votes
1answer
141 views

Torsion and Non-metricity Tensor on a Surface

In differential geometry of surfaces, how can one define a non-zero Torsion tensor? It seems that the connection you provide has always to be symmetric since, by definition, ...
4
votes
1answer
410 views

Laplace-Beltrami on a sphere

I'm trying to compute the Laplace-Beltrami of the function $u(r,\varphi,\theta) = 12\sin(3\varphi)\sin^3(\theta)$ on a unit sphere. Note that $\varphi$ is the azimuth, i.e. $\varphi \in [0,2\pi]$ and ...
1
vote
3answers
4k views

How to calculate surface area of a curved plane?

could anyone explain how to calculate the surface area of a curved plane? I am trying to calculate the surface area of a "vaulted" ceiling that is 24' long, 7' wide, and the height of the curve is 4' ...
1
vote
2answers
134 views

Find a point through which every surface tangent to z=xe^(y/x) passes

Find a point through which every plane tangent to the surface $$ z=xe^{\frac{y}{x}} $$ passes. It's not a homework. I know, that I need a normal vector and the point of tangency to find a tangent ...
1
vote
1answer
80 views

Identify the Euler characteristic of the edge word $ abc^{-1}b^{-1}da^{-1}d^{-1} c $

Identify the Euler characteristic of the edge word $ abc^{-1}b^{-1}da^{-1}d^{-1} c $. The Euler characteristic is $$ X=V-E+F$$ where $V$, $E$ and $F$ are the vertices, edges and faces ...
4
votes
0answers
373 views

The Birman–Hilden Theorem and the Nielsen–Thurston classification

So this post is half question/half reference request, as I'm sure it's the kind of thing people would have thought about before (and indeed the question might even be trivial), but I've been unable to ...
4
votes
1answer
104 views

How many types of surface singularities multiplicity two exist?

All varieties are over $\mathbb{C}$. Let $S$ be a reduced algebraic surface in $\mathbb{P}^3$ with a singular point $p$ of multiplicity two. The question is local so we reduce to $S \subset ...
1
vote
0answers
117 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 ...
14
votes
4answers
230 views

How to show in a clean way that $z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$ is a torus?

How to show in a clean way that the zero-locus of $$z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$$ is a torus?
5
votes
1answer
131 views

Abelian Elliptic Surfaces

By abelian surface we mean a 2-dimensional algebraic complex torus. Thus $$ S=\Bbb{C}^2/\Gamma$$ where $\Gamma$ is a rank $4$ lattice in $\Bbb{C}^2$ and such that $S$ is algebraic. It has trivial ...
0
votes
1answer
38 views

Find position on surface of a lens

If I have a lens with coordinates UV on the lens surface where U, V are [-1, 1] and I want to find the real-world (x,y,z) coordinates of the UV point, how would I do that if I have the following ...
4
votes
3answers
91 views

Good source to learn about surface singularities?

I am looking for something that treats singularities on algebraic surfaces and curves over $\mathbb{C}$, starting from the very basics but not stopping there. I checked out Miles Reid his lectures on ...
1
vote
1answer
60 views

Uniqueness of Seifert graphs

If we make the bands and disks of a Seifert surface really small and really thin the surface collapses to a graph. It is called a Seifert graph. If it is not a directed and weighted graph, can we ...
0
votes
4answers
436 views

Prove that Gauss map on M is surjective

Let $M$ be a closed, orientable, and bounded surface in $\mathbb{R}^3$. (a) Prove that the Gauss map on $M$ is surjective. (b) Let $K_+(p) = \max \{0, K(p)\}$. Show that $$ \int K_+dA \ge 4\pi. $$ ...
2
votes
1answer
62 views

Graphs from Seifert surfaces

Given a Seifert surface if we make the disks and bands infinitely small and thin it becomes a graph where the disks are vertices and the bands are edges. Can we say that following theorem, For ...
4
votes
0answers
76 views

Space of solutions to $f(x+y) = f(x) + f(y)$ when $f$ is convex

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as $\Omega_f(x) \triangleq ...
1
vote
2answers
145 views

Uniqueness of Seifert surfaces of knots

I know the theorem that Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K. So, can we also say that for every unique Seifert surface there is an unique knot and vice ...
1
vote
1answer
43 views

The continuity of principal coordinate system

$X$ is a $C^k$ hypersurface in $\mathbb R^{n+1}$ and $y$ is a fixed point on $X$. Can we find an orthogonal system $\{e_1(x),e_2(x),\cdots,e_{n+1}(x)\}$ on a neighborhood $U$ of $y$ such that 1. ...