For questions about surfaces.

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3
votes
1answer
403 views

is this set a regular surface?

I'm reading "Differential Geometry of Curves and Surfaces of Manfredo Docarmo" I'm doing the exercises of the chapter 2. Here is the definition of regular surface that we are following: I have ...
0
votes
1answer
103 views

Conics generalized to surfaces of constant curvature

Do conic sections have an interesting generalization to surfaces of constant curvature? Consider a sphere (constant positive curvature) $\mathcal{S}$ centered at $O$, as well as points $A, B \in ...
3
votes
2answers
133 views

A vanishing theorem for differential forms.

I am trying to prove that for an algebraic surface $X$ (under some extra assumptions that are probably not important) there the space $H^0(X,\Omega_X^1)$ is trivial, i.e. that there exist no globally ...
4
votes
1answer
65 views

Holomorphic 1-forms in $y^2-(z-a_1)\ldots(z-a_n)$

I know that the surface $y^2-(z-a_1)\ldots(z-a_n)$ is a Riemann Surface (that is the Riemann surface of $\sqrt{P(z)}$ with $P(z)=(z-a_1)\ldots(z-a_n)$) of genus $g$ and that ...
0
votes
1answer
216 views

Intersection of plane with quadratic regression surface

A quadratic OLS regression with two predictors is defined as: $Z = b_0 + b_1X + b_2Y + b_3X^2 + b_4XY + b_5Y^2 + e$ (1) If this regression surface is plotted, it can look like this ($b_0=10, ...
3
votes
1answer
54 views

Compact orientable surfaces embeddings

I wonder if we can embed a compact orientable surface of genus $g$ into another of genus $g'$, if $g < g'$. I already know that this is false if $g>g'$, because of the first homology groups. Any ...
1
vote
0answers
47 views

Are there moduli spaces of higher-dimensional varieties

In short, the answer to the question is yes. I'm aware of the existence of moduli spaces for canonically polarized varieties with fixed Hilbert polynomial over $\mathbf C$. I think they require the ...
2
votes
0answers
146 views

calculating the area on the surface os a sphere created by intersection of two spherical caps!

Consider a spherical object composed of two compartments (A and B, not necessarily hemispheres) sitting at the interface which is characterized by a plane separating 1 and 2. For this case, ...
1
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0answers
146 views

Is there a general equation expressing the area of a spherical quadrilateral using two polar angles, an azimuth, and a radius?

Is there a general equation expressing the area of a spherical quadrilateral using two polar angles, $\theta_1$ and $\theta_2$, an azimuth, $\phi$, and a radius, $r$? I know this can be done by ...
4
votes
2answers
218 views

Convoluting two surfaces

I'm wondering how the concept of convolution can be extended to 2D. As example, let us take a constant function $z=f(x,y)=1$ with support on $[0..1]^2 \in \mathbb{R}^2$ (see Fig. 1). If we now ...
0
votes
1answer
406 views

Help me understand a surface integral question?

The question is: Evaluate the surface integral: $$ \iint\limits_S \, x^2yz\ \mathrm{d} S $$ Where S is part of the plane z = 1 + 2x + 3y that lies above the rectangle [0,3] X [0,2] I literally just ...
3
votes
1answer
96 views

Reference request for the existence of triangulation of surfaces

I would like to read a self-contained proof of the theorem of Rado that states that any second countable topological surface admits a triangulation. Actually, I would be content with the compact case. ...
1
vote
2answers
651 views

What is the difference between surface area and scalar surface integrals?

What is the difference between the surface area of a paremetrized surface and the scalar surface integral of a function in $\mathbb{R}^3$? Are they not the same thing?
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 ...
1
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0answers
169 views

NURBS and subdivision surfaces

I need some links or advices on NURBS, T-Splines, subdivision surfaces and related stuffs. I try to evaluate possibilities for modeling and rendering surfaces (and curves). Subdivision surfaces and ...
1
vote
2answers
138 views

Surface of revolution using cylindrical polars

Consider the surface of revolution of the curve $$y = x^2$$ where $0 < x < 1$. By writing a suitable integral, show that the area of this surface is 3.81 units. (You are advised to work in ...
2
votes
1answer
198 views

Surfaces of revolution - Problem

Here is the problem: Let $S$ $\subset$ $\mathbb{R^{3}}$ be a regular surface with the property that all normal lines meet the z-axis. Prove that S is contained in a revolution surface around z. I ...
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 ...
1
vote
1answer
66 views

Virtual knot diagrams on surfaces with genus?

To the best of my limited understanding, a virtual knot diagram may be thought of as the projection of an embedding of $\mathbb{S}^1$ in a 2-manifold with genus onto $\mathbb{R}^2$. That is to say it ...
1
vote
1answer
475 views

Surface Area Problem with Specific Formulas

I noticed this question in a Math Problem-Set book. These were the only formulas allowed: $1. Area(quadrant) = \frac{1}{4}\pi r^2$ $2. Area(square) = (side)^2$$3.Area(semicircle) = \frac{1}{2} \pi ...
4
votes
1answer
186 views

Why does it appear that Willmore energy is always zero?

The answer is "because I'm being sloppy," but the problem is I don't know exactly where I'm being sloppy. Here's my sloppy argument: Let $M$ be a smooth compact surface without boundary in ...
2
votes
4answers
1k views

Equation for distance from a point outside a sphere to any point on its surface

I have a point m outside a sphere. The sphere center is o and r is the radius of sphere. Distance from point m to o is l. If we draw a line from m to any point on the surface of sphere, this line has ...
2
votes
0answers
101 views

Do K3-surfaces have Weierstrass equations

I've been wondering a bit about K3-surfaces and their analogy to elliptic curves. I've just started so this might be a very silly question. Do all K3-surfaces have a Weierstrass equation (up to ...
0
votes
1answer
203 views

$S-\{p\}$ admits a bouquet of circles as deformation retract.

Let $S$ be a closed compact surface, $p\in S$ and $X=S-\{p\}$. Show that X admits a bouquet of circles as deformation retract. How many circles? I'm starting to study algebraic topology and I can't ...
6
votes
0answers
132 views

Normal subgroups of the fundamental group of a non-orientable surface.

Let $N^2_g$ be a non-orientable closed genus $g\geq 2$ surface. Is there a way to explicitly list the normal subgroups of $\pi_1(N^2_g)$ in terms of generators and relations? I am interested in ...
0
votes
1answer
43 views

Consequence of the compactness of a hyperbolic surface

As part of a course I'm taking this semester, I am studying surfaces from this book http://www.math.brown.edu/~res/Papers/surfacebook.pdf. On page 142, the author presents a proof of the fact that ...
1
vote
2answers
547 views

Euler characteristic of a surface

It is known that a closed orientable surface of genus $g$ has Euler characteristic $2-2g$. According to this, the open disc being of genus $0$ should have Euler characteristic $2$, but this ...
3
votes
0answers
69 views

Do we really need to use the Jordan-Schönflies Theorem to prove that every surface can be triangulated?

I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds ...
1
vote
1answer
292 views

Christoffel symbols in Differential geometry iff proof

I need help in proving that $H = 0$ for a surface iff $g_{11}L_{22} - 2g_{12}L_{12} + g_{22}L_{11} = 0.$ I think that these are the Christoffel symbols exploited in some manner and normally, I'm not ...
2
votes
1answer
213 views

Mathematical name for the horn shape

I am looking for the technical name for the horn shape which is created by repeating circles while increasing the radius size varying with an exponential function. Any references that can help me find ...
0
votes
1answer
98 views

How do you explain the appearance of a sine in the integral for calculating the surface area of a sphere? [duplicate]

Let's say I want to calculate the surface area of a sphere. For simplicity, let's just use the unit sphere. A naïve argument might go like this. Let's say I mark the north and south "poles" and draw ...
1
vote
0answers
46 views

Sign-preservation of continuous map in a small neighborhood

So I was reading a small book on surfaces called "Mostly Surfaces" which is available for free in the internet: http://www.math.brown.edu/~res/Papers/surfacebook.pdf In page 32, the author decides ...
1
vote
0answers
40 views

What is the meaning of a surface approximation equation?

Given a set of $n$ points $P$, a point $p_i\in{P}$, $1\leq i\leq n$ and a number $k<n$, I define the group $N_k(p_i)$ as the group containing $p_i$'s $k$ nearest neighbors. In addition, each point ...
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.
2
votes
1answer
159 views

understanding (and plotting) a surface (from implicit to parametric?) [duplicate]

Possible Duplicate: Automation of 3D Paper Modeling i am a programmer and not a mathematician, and my math knowledge are a little bit rusty. i have found this nice picture surfing on the ...
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 ...
4
votes
0answers
206 views

Morse theory and homology of an algebraic surface (example)

Let $T_n$ denote the $n$-th Chebyshev polynomial and define $f_n(x,y,z)\!:=\!T_n(x)\!+\!T_n(y)\!+\!T_n(z)$ and $$Z_n:=\mathcal{Z}(f_n) \subseteq \mathbb{R}^3,$$ the Bachoff-Chmutov surface, where in ...
3
votes
1answer
264 views

Computing the homology of a torus by relative homology of a cylinder

I was trying to compute the homology of a torus with the long exact sequence for relative homology formed quotient, inclusion, and boundary $ \dots\to\tilde{H}_n(A)\overset{i_*}{\to} ...
1
vote
1answer
933 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, ...
0
votes
1answer
125 views

Surface integral

Without getting into the whole question, I was asked to evaluate a surface integral $\iint\limits_S f(x,y,z) da$ where S is the cylinder $x^2 + y^2 = x$ between $z=a$ and $z=b$ Now normally I ...
1
vote
2answers
144 views

Transform flat surface into paraboloid

Is it possible to transform a flat surface into a paraboloid $$z=x^2+y^2$$ such that there is no strain in the circular in the circular cross section (direction vector A)? If the answer is yes, is ...
0
votes
2answers
775 views

Finding a surface fitting equation for this set of data

I have a question regarding surface fitting (3D curve fitting), and since I am not from a maths/stats background I was wondering if someone can help me or point me to the right resources? I had ...
5
votes
2answers
347 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.
0
votes
1answer
66 views

practical question about developable surface

there's my question: Given 2 regular plane curves (let's say $\mathcal{C}^1$) in the 3D space, is there always a developable surface which contains both curves ? Thanks, anders
6
votes
1answer
105 views

What surfaces in $\mathbb R^3$ are such that every planar section (with more than 1 point) has nontrivial symmetry?

In $\mathbb R^3$ , the intersection of a plane and a sphere (e.g. $x^2 + y^2 + z^2 = 1$) is either empty, a single point, or a circle. All isometries of those circles are realized by isometries of ...
0
votes
2answers
111 views

Function of the surface obtained by rotating the graph of $\frac{1}{|x|}$

What is the function of the 3 dimensional plane created when the graph of 1/abs(x) is rotated in the z-axis around the origin? I'm sorry for bad formatting and if this is a duplicate.
1
vote
2answers
474 views

Given a vertex and the base curve, how to find the equation of a cone [duplicate]

Possible Duplicate: Can any smooth planar curve which is closed, be a base for a 3 dimensional cone? Lets say a vertex V is given as $(\alpha ,\beta ,\gamma )$ and the base of the cone is ...
1
vote
1answer
81 views

How do I find the number of vertices on a planar diagram of a surface?

Cheers, I have a question which I just do not seem to see the answer for: I am proving the classification theorem for compact surfaces and use planar diagrams as representation of the surfaces. I ...
1
vote
1answer
201 views

Complete developable surface in $\mathbb{R}^3$ is ruled

Let $X \subset \mathbb{R}^3$ be a complete smooth surface which is developable in the sense that its Gaussian curvature is identically zero. Wikipedia claims that such a surface is necessarily ruled, ...
1
vote
1answer
827 views

Quadric surface graphing applet

Does anybody know of any online tool/applet that can be used to graph quadric surfaces? i.e. If I want an elliptic paraboloid, I can click on "elliptic paraboloid" and enter my own specified values ...