For questions about surfaces.

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3
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1answer
52 views

How do we check conformal equivalence of parametrized surfaces, e.g. parallel surfaces?

Suppose we have two parametrized surfaces in $\mathbb{R}^3$: $$ X,Y:\mathbb{R}^2 \rightarrow \mathbb{R}^3 $$ The induced metric on either surface is the pullback of the Euclidean metric $\bar g$ due ...
0
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1answer
44 views

Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
1
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1answer
42 views

Where can I find a good set of notes discussing main theorems/ideas surrounding non-orientable surfaces?

I'm currently looking at non-orientable surfaces, but know very little about them. Is there are good set of notes that will teach me the classical results surrounding non-orientable surfaces?
0
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1answer
77 views

Surface Integral calc 3

I am having difficulty setting up this problem. I know the bounds must be 0 to pi/2 for both theta and phi but I am unsure as to how to calculate the integrand. I know it must be the double integral ...
1
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2answers
123 views

Representation of nullhomologous loop on compact surface as a product of commutators.

Why this sentence is true?: Assume that $M$ is compact surface and $f: S^1 \to M$ is nullhomologous and without selfintersections. Letting $g$ be the genus and $b$ the number of boundary components ...
2
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0answers
31 views

common surface between two equation

what is common surface between: $(x+5)^2+z^2=y$ and $z^2+y^2=25$ ? I have found that at the XY plane the common surface is hiperbola, but it cannot be right because at the paraboloid there aren't any ...
0
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1answer
178 views

Surface fitting

I do not need a complete answer but just some advice. I have a sparse matrix of points in a volume. I know a surface passing by these points exists and this surface is mostly flat and relatively ...
0
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1answer
141 views

Need function for 2D sigmoid-shaped monotonic Surface

I am looking for a 2D function, $f(x, y)$ which increases monotonically over the range $(0,0)$ to $(1,1)$. In other words, it will be $0$ at $(0,0)$ and $1$ at $(1,1)$. It will also evaluate to $0$ ...
0
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1answer
103 views

Building bicubic coons patch from four boundary curves

I want to create s coons patch surface from four boundary curves s1(u), s2(u) q1(v), q2(v) I know that equations are the following (added screenshots from a presentation): There are a few ...
0
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1answer
57 views

Linear equivalence vs algebraic equivalence of divisors on smooth projective surfaces

Let $X$ be a smooth projective surface and $D_1, D_2$ be two divisors on $X$. Is it true that $D_1$ is linearly equivalent to $D_2$ if and only if $D_1$ is algebraically equivalent to $D_2$?
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0answers
33 views

ample anticanonical system implies regularity

I was having a look at Exercise V.21(1) of Beauville's "Complex Algebraic Surfaces", where it is asked to classify surfaces with ample anticanonical system. These are the surface $\mathbb{P}^1 \times ...
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0answers
84 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
2
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1answer
50 views

Creating a surface from a path of 3D cubic bezier curves

I have a list of cubic bezier curves in 3D, such that the curves are connected to each other and closes a cycle. I am looking for a way to create a surface from the bezier curves. Eventually i want ...
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0answers
31 views

How do points change in a curved surface?

In the middle picture it shows a row of sticks at certain points along a flat surface. Now in the outer left picture (never-mind the outer right one), when the surface becomes curved the points ...
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2answers
292 views

Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
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0answers
74 views

NURBS surface fitting for a closed region on mesh

I'm developing a tool that allows users to select a closed boundary (a polygon) on the triangle mesh and then from this boundary, generate a NURBS surface fitting the original mesh surface. My idea ...
-2
votes
1answer
83 views

Cw complex $\Sigma_g$

Consider the oriented connected compact surface $\Sigma_g$ of genus $g$ with its standard CW structure. How do I write down the attaching map for the single $2$-cell and how can it be proven that it ...
3
votes
1answer
68 views

For compact surface $M$ and loop $f$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such that $f \notin \ker(\phi)$

Why is this sentence true? For every not nullhomologous loop $f$ without selfintersections on orientable compact surface $M$ there exists a surjection $\phi: H_1(M) \to \mathbb{Z}/8\mathbb{Z}$ such ...
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2answers
40 views

Approximate a surface by set of points

Given a set of points $(X, Y, Z)$ obtained from the experimental data that can be considered as a 3D surface. What is the common approach to get an approximating function Z=f(x,y) that describes the ...
1
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1answer
101 views

How do you find the surface area of a boundary in $\mathbb{R}^3$?

I need to solve this problem: Let $D=\{(x,y,z):4(x-2+z)^2+4y^2\le(2-z)^2,0\le x-z\le1\}$ Calculate the area of $\partial D$ So how do you calculate the area of the boundary of a volume defined ...
3
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1answer
50 views

Does a pseudo-Anosov homeomorphism of a punctured surface possess infinitely many periodic points?

In A Primer on Mapping Class Groups by Farb and Margalit theorem 14.19 implies that every pseudo-Anosov homeomorphism $f:S \rightarrow S$ on a compact surface $S$ possesses infinitely many periodic ...
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0answers
35 views

How to find the tangents for a spiral geometry at each point accurately

I have been working on a progam that generates spirals from contours that have been formed by slicing a surface by various planes along its height.The contours are a collection of linear line ...
0
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1answer
20 views

tilt of surface from the normals

I have a flat object (not totally flat (let's say in range of 25µm)) which I measured two times (The measuring concept is not important here) with applying a tilt between the two times. I have the ...
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0answers
46 views

Difficult Surface Integral

I am trying to perform a surface integral over kind of a weird shape. So the radius of the shape should be equal to the multiple of $3$ constants (one for each of the $x, y$ and $z$ directions) each ...
1
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1answer
84 views

Parallel Transport - Path independence

I'm trying to solve this problem: Prove that if the parallel transport is path independent, i.e., given two points $p,q \in S$ the parallel transport from $p$ to $q$ is the same, no matter the curve ...
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4answers
68 views

Help calculating the surface area given by the polar curve: $r=2(1-\cos\theta)$

I want to calculate the surface area given by the curve: $$ r = 2(1-\cos(\theta)) $$ using an integral. I have thought about doing this: $$ x = r\cos(\theta), \, y = r\sin(\theta) $$ $$ \iint r \,dr ...
2
votes
1answer
31 views

Newtonian potential at (0, 0, – a)

I found this problem in the book Advanced Calculus, written by Friedman. "Newtonian potential at (0, 0, – a) due to a mass with constant densinty $\sigma$ on the hemisphere S: $x^2 + y^2 + z^2 = ...
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1answer
38 views

Topology of level surfaces

I have a level surface of the form $f(x,y,z,w)=0$ and also $g(x,y,z)=0$. Here f and g are differentiable! I need to decide if they are compact or not. Is there any criteria, theorem or anything? ...
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0answers
42 views

Blowing up a Singular Point More Than Once.

I am trying to understand how $I_n$-fibres appear in an elliptic surface by performing a sequence of blow-ups. To be concrete, I am looking at the following elliptic surface given in Weierstrass ...
1
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1answer
36 views

Is $\textrm{im}(f)$ homeomorphic to the torus less the inner equator?

Consider the map $id_{S^1}\times f:S^1\times [0, 1]\longrightarrow S^1\times S^1$ where $f:[0, 1]\longrightarrow S^1$ is given by $$f(t)=(\cos(\pi t), \sin(\pi t)).$$ Is it true that $\textrm{im}(f)$ ...
2
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2answers
89 views

Regular Surface: Regularity Condition

I am having some difficulty in understanding the meaning/motivation of the regularity condition in the definition of regular surfaces. The definition (restricted to $\mathbb{R}^2$ and ...
2
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2answers
111 views

Formula for a surface of revolution

The curve $y=\sqrt{x^2+1}, 0\leqslant{x}\leqslant{\sqrt{2}}$, which is part of the upper branch of the hyperbola $y^2-x^2=1$, is revolved about x-axis to generate a surface. Find the area of the ...
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0answers
15 views

Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is: ...
0
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1answer
42 views

Understanding surface integrals

My question is a bit vague, but I'm trying to get a better understanding of surface integrals and their relation to physics. Suppose I have a surface, say a sphere, and I have a function which gives ...
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2answers
32 views

Surface: intersection of 2 polar curves

I have these two polar curves: $$ C_1: r = 2 - \cos(\theta)\\ C_2: r = 3 \cos(\theta) $$ Plots: C1 and C2. I need to find the surface of $D = C_1 \cap C_2$. I started by finding the solution to ...
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3answers
73 views

surface integral using substitution

I am stuck trying to calculate the following surface integral: $$\int _{R}\int (x+y)^{2}ds$$ over the the following regions: $$0\leqslant x+2y\leqslant 2\: \: \wedge \: \: 0\leqslant x-y\leqslant ...
0
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0answers
17 views

Bezier Surface evalution

So the problem I'm having at the moment, is a thinking problem. I can draw a bezier surface (parametric surface) with 16 control points and if I evaluate S(u, v) I get a coordinate in the 3D space. ...
1
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1answer
69 views

the fundamental group of punctured surface

Let $S_{g,m}$ be a surface of genus $g$ with $m$ punctured, we know the fundamental group of $S_{g,0}$ is $$ \pi_1(S_{g,0}) = \left\langle a_1, b_1, \dots, a_g, b_g {~\large\mid~} [a_1, b_1] \dots ...
0
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2answers
66 views

Better way to denote position on a sphere's surface

TL;DR: Read the bold text. If you have a rectangular plane, you can use two coordinates (X, Y) to define any position on the plane. If you have a sphere, you can still use polar coordinates to denote ...
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0answers
102 views

Best book for learning multiple integrals, line integrals, greens theorem etc..

I've been searching for a book that teaches multiple integrals and such in a way that I can understand, I need to learn it quickly, so I don't need too much of the intuition, I just need to be able to ...
0
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1answer
47 views

Surface area of sphere in N dimensions; and a failed extension to ellipsoids

I'll present a calculation of the surface area of a sphere in $N$-dimensions. This calculation is performed in cartesian coordinates. I haven't seen the computation done this way before (though I ...
3
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1answer
55 views

Surface area of the part of the sphere $x^2+y^2+z^2=a^2$ that is inside the cylinder $x^2+y^2=ax$

I've been solving some surface area problems lately, but I don't think that the same approach that I was using will work with this one (or at least will result in a lot work). So, I believe I should ...
4
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1answer
212 views

Metric Tensors and its Taylor Expansion in Normal Coordinates

With metric tensors of the unit sphere in normal coordinates, their Taylor series for $p\in S$ near the north pole $N$ can be written as follows. $$g_{rr}(p) \equiv 1; g_{r\theta}(p) = g_{\theta ...
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1answer
34 views

Name of the surface with two sides and three boundaries

Once i have seen a 3d visualization of a surface with the following characteristics: it had three circular borders. If you imagine the surface inscribed in the earth globe, one of the borders would ...
4
votes
2answers
65 views

What curvature conditions make a surface rigid?

Consider a compact surface $S$, possibly with boundary, embedded in $\mathbb{R}^3$, with the induced Riemannian metric. I believe that if $S$ has constant positive Gaussian curvature (that is, $S$ is ...
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0answers
87 views

How to estimate/determine surface normals and tangent planes at points of a depth image (point cloud)?

I have depth image, that I've generated using 3D CAD data. This depth image can also be taken from a depth imaging sensor such as Kinect or a stereo camera. So basically it is depth map of points ...
0
votes
1answer
32 views

Local diffeoorphism and orientability of surfaces

I need some help to prove this: Let $S_2$ be an orientable regular surface and $f : S1 \rightarrow S2$ be a local diff eomorphism. Then $S_1$ is an orientable surface. Thanks.
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0answers
36 views

the fundamental group acts on half upper plan

Let $S$ be a compact oriental surface without boundary of genus $g\ge 2$, then its universal covering is $\mathbb{H}^2$, I am confused with 2 facts following: (1) $\rho:\pi_1(S)\hookrightarrow ...
3
votes
1answer
99 views

Homology subgroups generated by non-intersecting cycles

Suppose I have a closed genus $g$ surface. I can pick a canonical homology basis for the surface by picking $g$ "A-cycles" $a_1,\ldots,a_g$, and then $g$ "B-cycles" $b_1,\ldots,b_g$, represented by ...
1
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1answer
40 views

Surface described by parametric equations

If I've got the surface in $\mathbb{R}^3$ described by: $x(s,t)=s^2-t^2$, $y(s,t)=s+t$, $z(s,t)=s^2+3t$ for $(s,t)\in\mathbb{R}^2$, and I'm told this surface is the graph of a function $f(x,y)$, how ...