For questions about surfaces.

learn more… | top users | synonyms

3
votes
0answers
43 views

Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside a cylinder.

(a) Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside the region $\frac{x^2}{M^2}+\frac{y^2}{N^2}=1$ (b) Find the surface integral of $f=|xy|$ over ...
0
votes
1answer
228 views

Outer Unit Normal: Cylinder

I have a cylinder occupying the region $x_{1}^{2}+x_{2}^{2} = R^2$ and $-G< x_3 < 0$ All I want to do is define the outer unit normal on the curved face. I thought about just calling it $e_1$ ...
0
votes
1answer
37 views

Is there a definition of cylinder that these equations satisfy

Our teacher is claiming that (in $\mathbb{R}^3$) the following surfaces are "cylinders": $3x+y+\frac{7}{2}=0$ $y=x^2$ $z^2 = y$ $\frac{x^2}{4} + \frac{y^2}{4} = 1$ Is there any definition of ...
4
votes
1answer
109 views

Trouble computing the shape operator.

Where have I gone wrong in the following computation of the shape operator of surface? Suppose we have a surface $M = \{(x,y,f(x,y)) \: | \: (x,y) \in \mathbb{R}^2 \}$ for some nice ...
2
votes
1answer
110 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 ...
0
votes
1answer
85 views

Parametric surfaces - Parameterization of torus

A rotational surface area is created when a curve in the $xz$-plane, with parameterization $\def\i{\pmb{i}}\def\k{\pmb k}$ $r=x(t)\i + z(t)\k$ , $t \in [a,b]$, rotates around the $z$-axis. This ...
0
votes
2answers
83 views

surface vs differentiable manifold

Every surface is a smooth manifold, but the reciprocal is verified? some concrete example of a differentiable manifold is not surface? Thanks in advance for the suggestions.
2
votes
1answer
95 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?
1
vote
0answers
44 views

Gauss and Stocks teory

Given $\phi\in C^1(R)$, and we define the curve and surface $\gamma=${$(x,y):y=\phi(x),0\le x\le 1$} $S=${$(x,y,z):z=\phi(\sqrt{x^2+y^2}),x^2+y^2\le 1$} a.I need to prove that $A(S)=2\pi\int_\gamma ...
0
votes
1answer
67 views

Surface equation for a triangle when vertices are given

How to find equation for surface of a triangle when vertices are given? Such as when vertices are $(1,0,0),(0,1,0),(0,0,1)$. Surface given by $x+y+z=1$.
2
votes
1answer
59 views

Radial geodesics in a graph of a function

I'm trying to figure out how to prove the following claim: Suppose that $S$ is the graph of a function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and every plane containing the $z$-axis intersects $S$ ...
2
votes
1answer
75 views

Whitney umbrella birational to $\mathbb{A}^2$ but not isomorphic

Define the Whitney umbrella as the affine surface $V(z^2 - yx^2) \subset \mathbb{A}^3$. I've come across an exercise that asks me to show that this surface is birational, but not isomorphic, to ...
1
vote
1answer
89 views

Difference between free homotopy and isotopy. Numer of non-isotopic curves.

I realize that whenever I think of two simple closed curves in a surface being isotopic I actually think of them as being freely homotopic (intuitively). I am really confused now. So I have the ...
1
vote
0answers
88 views

Surface orientation

Let $S_{1}$ and $S_{2}$ be two oriented surfaces ($N_{1}$ and $N_{2}$ their normal fields, respectively). We say that a local diffeomorphism $f$ : $S_{1}$$\rightarrow$$S_{2}$ preserves orientation if ...
0
votes
1answer
36 views

Search for sharp maximums of 4D surface

I need to find sharp local maximums of numerically defined 4D surface. I have a surface with lots of maximums. I already know how to find them all. Some of them look like this: wide extremum, others ...
3
votes
1answer
46 views

What does it mean for a surface to evolve with divergence-free velocity?

Suppose we have an evolving hypersurface which evolves with a velocity field $V$, such that $\nabla_S \cdot V = 0$ where $\nabla_S$ is the surface or tangential gradient. What does this mean? What ...
6
votes
0answers
100 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 ...
2
votes
0answers
111 views

Surface infinitesimals and its intuitive manipulation?

The excess pressure in the concave side of any liquid bubble or drop with surface tension of the liquid being $T$ is $\frac {4T}r$ and $\frac {2T}r$ respectively. I wanted to derive it using a ...
2
votes
0answers
75 views

Working with projection of areas?

I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting ...
1
vote
1answer
58 views

The equation of a surface created by the extrusion of a 2D closed curve along a path

How do I obtain the equation of a surface created by the extrusion of a circle (or ellipse) created on the XY plane along a parabola or a parametric curve which lies on the YZ plane. The goal is to ...
1
vote
2answers
34 views

Boy surface parameterization confusion

I'm looking at equations 10, 11 and 12 here. What do the letters I and R represent in these equations?
2
votes
1answer
112 views

Flowlines of blobs

I have the following formula for blobs/metaballs, which is said to be the same as the one used for electromagnetism: \begin{gather} f(x,y,z) = \frac{d(A,B)}{\sqrt{(x-xA)^2 + (y-yA)^2 + (z-zA)^2}} \\ ...
0
votes
0answers
52 views

Rheotomic surfaces parameterization?

Are there parameterizations for rheotomic surfaces? Or, am I stuck with implicit formulas and marching cubes for plotting points? Are there special cases where the surfaces are parameterizable? Here ...
0
votes
1answer
62 views

Find the area/surface of a figure, specified by inequalities

I have some difficulties in solving this problem: Find the area of a figure, specified by the inequalities: $x^2 + y^2 \leq 2x$ and $x^2 + 2x + y^2 \leq 3$ I know that I have to use the formula ...
0
votes
1answer
137 views

Numerical computation of surface curvature

In 2 dimensions, the definition of curvature of a curve $y = y(x)$ is \begin{equation} C = \frac{y''}{(1+y'^{2})^{3/2}} \end{equation} and it is easy to estimate the curvature numerically for given ...
3
votes
1answer
91 views

Euler characteristic for non-convex polyhedra.

I read a little introductory book to topology. It basically said that for any two-dimensional manifold (well maybe just the closed ones, as I think about it) its topology can be unambiguously defined ...
1
vote
1answer
25 views

Non-separating surfaces and the 2nd homology of a $3-$manifold

I have a question about the relation between non-separating surfaces and the nontrivial 2nd homology of a 3-manifold. The question is: Let $\Sigma$ be a closed nonseparating surface embedded in a ...
1
vote
0answers
83 views

Transversal and complete intersection of hypersurfaces in $\mathbb{P}^{n}$

(a) Let $k<n$ and $F_{1},\dots,F_{k}$ be homogeneous polynomials of degrees $d_{1},\dots,d_{k}$ of $n+1$ variables in generic case. Prove that the corresponding hypersurfaces in ...
1
vote
1answer
217 views

Restriction of a differentiable map $R^3\rightarrow R^3$ to a regular surface is also differentiable.

This is again an excercise from Do Carmo's book. Prove: if $f:R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$ is a regular surface invariant under $L,$ i.e, $L(S)\subset S$, then the ...
1
vote
1answer
42 views

Smooth grid lines and implicit surfaces

Take for instance the computer rendering of this implicit surface: http://xahlee.info/surface/cayley_cubic/cayley_cubic.html The image shows a grid on the surface. How do I calculate the connection ...
0
votes
1answer
87 views

Vectorfields on Surfaces with MuPAD

i am working with MuPAD. I can make vectorfields in 3D and also Surfaces in 3D via plot::VectorField3D and plot::Surface. But now i want to draw a vectorfield ON a surface. If $X$ is a vectorfield ...
0
votes
1answer
44 views

Get random position on surface

I'd like to get a random position on a surface of an object, and also follow it's normals. Example, let's say I have a sphere, I can get all the face, normal and vertex positions and well as their ...
1
vote
2answers
65 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
144 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 ...
5
votes
3answers
219 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
494 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
1answer
108 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
vote
0answers
154 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
67 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
98 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 ...
13
votes
2answers
195 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
91 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
34 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
125 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 ...
0
votes
0answers
38 views

Surface Reconstruction from Hessian Field

I am looking for references regarding surface reconstruction. Consider a point cloud in $\mathbb{R}^3$ with the Hessian (or possibly second fundamental form) defined at each point. I would like to ...
1
vote
1answer
99 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
304 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 ...
10
votes
1answer
144 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
248 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
44 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 ...