For questions about surfaces.

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2
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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
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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
200 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
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0answers
128 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
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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
520 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
68 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
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1answer
290 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
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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
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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
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0answers
44 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 ...
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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
119 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
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1answer
158 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
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2answers
132 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
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0answers
204 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
257 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
908 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
122 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
139 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
724 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
346 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
65 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
108 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
453 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
199 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
818 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 ...
2
votes
1answer
155 views

Drying blood - an algorithm for calculating the geometry of blood stains

Motivation A bucket full of blood gets spilled over the floor. Question: What shape will the dried blood stains have? Abstraction The blood is modeled by a set of interacting particles (e.g. SPH). ...
3
votes
4answers
372 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
215 views

Give the equation of the surface

Given $$z = y^2 + 3,$$ give the equation of the surface if rotated around the $z$-axis. After I plot this out, I get a simple parabola in the $yz$-plane... so flipping it about the $z$-axis is just a ...
0
votes
1answer
112 views

great circle distance

in the euclidean plane the distance from the origin to a point is $s^2 = x^2 + y^2 $ I am reading a paper which say that this could be called an algabraic metric for the plane. the paper then ...
3
votes
1answer
290 views

Rank of first homology group for surface with punctures?

I feel like this question will be a head-slapper once I figure out the answer, but for the moment I'm having trouble! Let $M$ be a compact, connected, orientable 2-manifold of genus $g$ with $b$ ...
5
votes
2answers
1k views

implicit equation for “double torus” (genus 2 orientable surface)

The embedded torus in $\mathbb R^3$ can be described by the set of points in $(x,y,z)\in \mathbb R^3$ satisfying $T(x,y,z)=0$, where $T$ is the polynomial ...
3
votes
2answers
26 views

Boundedness of Surfaces in $\mathbb R^3$

GIven an equation such as $ax^2+by^2+cz^2+dxy+exz+fyz=g$ where $a,b,c,d,e,f,g\in \mathbb R$, How can we tell if the surface described is a bounded one without explicitly plotting a graph?
0
votes
1answer
136 views

How to draw a cone in Sage?

Given a cone with equation z^2 = x^2 + y^2, how would I draw it in Sage? I tried turning it into a function and passing arguments but it didn't work out for me.
1
vote
2answers
93 views

smooth K3 surface

In his paper "Examples of Calabi-Yau 3-manifolds with complex multiplication", Jan Christian Rohdes claims that the surface $S \subset \mathbb{P}^3$, with variables $(y_2: y_1: x_1: x_0)$, given by ...
2
votes
2answers
333 views

projection of a quadric surface

Consider the quadric surface $X = \{ xy = zw \} \subset \mathbb{P}^3$ and pick a point $x \in X$. I think it is true that if we think of $\mathbb{P}^2$ as the space of lines through $x$ in ...
6
votes
2answers
332 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 ...
1
vote
1answer
336 views

Estimate the surface area of a 2D shape where the only known value is the length of the enclosing boundary

Wondering if it is possible to estimate the surface area of a 2D shape where the only known value is the length of the enclosing boundary, and that it is know the internal surface area is solid. ...
4
votes
1answer
292 views

Del Pezzo surface of degree 4 is intersection of two quadrics?

Let $S$ be a del Pezzo surface $S$ of degree $4$. There is an exact sequence $$ 0\to H^0(\mathbb{P}^4,I_S(2)) \to H^0(\mathbb{P}^4,\mathcal{O}(2))\to H^0(S,\mathcal{O}_S(2))\to0$$ where $I_S$ is the ...
0
votes
2answers
236 views

How to parameterize a hyperboloid in a solid of revolution

The middle “hyperboloid” part of the solid of revolution is determined entirely by a single edge of the cube that does not touch one of the axis vertices - there are six such edges. Mark these ...
1
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1answer
187 views

Shortest distance to a surface

Let $S$ be a surface in $\mathbb{R}^3$ which is locally defined by a level set of some smooth function. Let $M$ be a point which is not on the surface. First of all, I would like to show that there ...
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1answer
194 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 ...
4
votes
0answers
203 views

Understanding surface area of a revolution/length of curve

I don't quite understand why the formula to find the surface area of a revolution is what it is: $$A = 2\pi \int_a^b x\ \sqrt{1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2}\ \text{d} x.$$ I ...
2
votes
1answer
175 views

Self-intersection of parametric surface using Gauss-Bonnet theorem

I am trying to detect when a closed parametric surface intersect itself. My surface is described as a triplet of parametric functions $x(u,v)$, $y(u,v)$ and $z(u,v)$ where $u,v\in[0,1]$. For that ...
2
votes
1answer
1k views

Shape operator vs second fundamental form

Is the any difference between shape operator and second fundamental form for surfaces?
2
votes
2answers
2k views

How to find surface area of $x=\sqrt{a^2-y^2}$

I still hard time to find surface area of function... I have The given curve is rotated about the $y$-axis. Find the area of the resulting surface. $$x= \sqrt{a^2-y^2},\quad ...
1
vote
2answers
802 views

Surface area formula

I'm kind of confused about the explanation of the surface area formula in my text book The text gave us $$\int_{a}^{b}2\pi f(x) \sqrt{1+[f'(x)]^2}dx$$ after that the formula is getting like ...