For questions about surfaces.

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3
votes
1answer
254 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
887 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
137 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
673 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
345 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
64 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
104 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
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2answers
437 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
198 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
811 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
153 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
370 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
213 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
286 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
135 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
332 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
328 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
333 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
291 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
234 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
vote
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 ...
1
vote
1answer
188 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
202 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
174 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
792 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 ...
2
votes
1answer
521 views

Parametric Equations for a $2$-torus

I know that for a torus (with one hole) the parametric equations describing it are $x= (c + a\cos v)\cos u, y= (c + a\cos v)\sin u, z= a\sin v$, where $c$ is the radius from the center of the hole to ...
3
votes
1answer
157 views

triangulation of pair of pants

How can we triangulate a pair of pants in a simple way? I am looking for some triangulation where I can compute the Euler characterstic easily (which is -1 for a pair of pants).
0
votes
1answer
82 views

surfaces of spheres are made of

Sorry in advance for lacking the appropriate terminology, please help me edit it in below. Take thease basic shapes: ...
1
vote
2answers
602 views

Meridian curves of surfaces of revolution are geodesics

Could someone explain how to go about proving that the meridian curves on a surface of revolution are geodesics?
2
votes
1answer
104 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 $X\times X$ ...
3
votes
2answers
85 views

Group of automorphisms of an orientable surface

If we consider the group of automorphisms of an orientable surface, then the subgroup that contains the orientation-preserving automorphisms will be of index two. Why is that? Any explanation will ...
5
votes
1answer
200 views

How to determine surface from given normal vectors and their distance on that surface

Situation: We have a bendable, non-stretchable surface, like a piece of cloth, with a regular grid on it. Unknown manipulation of the surface is done while preserving it's structure We recieve 3 ...
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vote
0answers
276 views

Understanding tensor product surfaces

Recently I bought a book about curves and surfaces (Curves and Surfaces for Computer Graphics by David Salomon), and I'm having trouble understanding the tensor product surfaces. I think I understand ...
2
votes
1answer
370 views

How to plot 3D figures correctly?

When we always draw e.g. cylinder on the whiteboard, we get kind of this: But naturally (assumed axes are perpendicular) when we have z-axis to the top and y-axis to the right, x-axis must point to ...
7
votes
1answer
148 views

Why can all surfaces with boundary be realized in $\mathbb{R}^3$?

I'm having trouble comprehending an informal proof of the fact that all compact surfaces with boundary can be realized in $\mathbb{R}^3$. I'm trying to find a proof of it on the internet, but I can't ...
2
votes
6answers
507 views

Software to display 3D surfaces

What are some examples of software or online services that can display surfaces that are defined implicitly (for example, the sphere $x^2 + y^2 + z^2 = 1$)? Please add an example of usage (if not ...
5
votes
0answers
81 views

surface of a torus by integration [duplicate]

Possible Duplicate: surface area of torus of revolution Let $R>r>0$ fixed. I want to compute the Area of $S=\operatorname{Im} \phi$ given by $$\phi(s,t):= \begin{pmatrix}(R+r\cos s ...
1
vote
2answers
125 views

can singular points become nonsingular after a base change

Let $X$ be a normal surface over a field $k$. Assume that $X$ is singular. Does there exist a field extension $L/k$ (finite or infinite) such that $X_L$ is nonsingular? The answer is no in general. ...
0
votes
1answer
254 views

Questions on surfaces touching along a curve (why is it a curvature line?)

I have two questions on surfaces touching along a curve. I would really appreciate if you could help me. i) Prove that if two surfaces touch along a curve and it is a curvature line on one of the ...
5
votes
1answer
379 views

Universality of Tate-conjectures

We all know that Prof.John Tate proposed a set of conjectures(along with Prof.Emil Artin) formally spread under the name of "Tate conjectures", they have a wide range of influence on various fields of ...
2
votes
1answer
105 views

The classification of surfaces

Can we completely classify the simply-connected surfaces (with or without boundary) in $\mathbb R^3$ up to homeomorphism?