For questions about surfaces.

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1answer
29 views

Is an open disk complete?

See the definitionS of complete surface, first definition: without edges. second definition: Any line segment can be continued indefinitely. By the first, open disk seems to be a complete surface, but ...
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1answer
41 views

Slice an ellipsoid into equally thick slices for maximal surface

After seeing a colleague slicing a nearly ellipsoid piece of ginger for his cup of tea into almost equally thick slices to get more surface area (so the tea would suck out the ginger taste better), i ...
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54 views

Basic question: degree of normal bundle is not self-intersection number

For $C$ a (possibly singular) curve on a nonsingular projective surface $X$, let's define $C^2=deg_C(\mathcal{O}_X(C))$. Why is it not the same as $deg_C(N_{X|C})$ when $C$ is singular? Why do ...
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1answer
178 views

How to get ellipse cross-section of an ellipsoid

I'm trying to get the major and minor radius of an ellipse which represents the cross-section of a given ellipsoid. This is particularly of interest in the field of RF propagation in terms of Fresnel ...
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1answer
35 views

Surface integrals where normal changes?

I am having problems getting my head around this problem: Evaluate the surface integral $$\int_S \vec F\bullet d\vec s$$ where $\vec F=x \vec i-y \vec j +z \vec k$ and where the surface S is ...
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15 views

Calculate area of subsection of a three-dimensional surface

For the following three-dimensional surface, z = -4.76 + 2.78x + 2.97y - 1.18xy, I would like to calculate the area (or percent of total area) for each of three subsections of this surface: (1) for ...
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1answer
36 views

Classification of Triangulated Surface

this is for a homework problem, although not the problem itself, and I'm looking for a little guidance. In the problem, I am given a very long list of triangles, approximately 40, and asked to ...
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1answer
84 views

Find the equation of the plane through a point which is perpendicular to a curve

Find the equation of the plane through the point $(1, -1, 2)$ which is perpendicular to the curve of intersection of the two surfaces $x^2 + y^2 - z = 0$ and $2x^2 + 3y^2 + z^2 - 9 = 0$. And would ...
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1answer
53 views

Think of the surface of genus $k$ as a sphere with $k$ tubes sewn in. Calculate its Euler characteristic by trangulating.

Think of the surface of genus $k$ as a sphere with $k$ tubes sewn in. Calculate its Euler characteristic by trangulating. I know that I need to make the genus covered by infinitely many triangle then ...
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1answer
45 views

Derive the formula for Gaussian curvature of the surface $z(x,y)$

Now I am assuming that this problem is referring to a Monge patch i.e. $\sigma(x,y) = (x,y,z(x,y)).$ I know the Gaussian curvature of a Monge patch can be rewritten as $$k = \frac{z_{xx}z_{yy} - ...
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52 views

Second fundamental form

Is it correct to say that .. the second fundamental form of surface theory determines the Euler characteristic and the genus of the surface ? If not how is it determined?
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1answer
41 views

Surface Area of the top half of an Astroid

How would I go about beginning this question? I have applied the standard surface area integral formula but it becomes complicated quickly.
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1answer
56 views

Asymptotic lines

I have a surface $f : \Omega \rightarrow \mathbb{R}^3$ that is represented by $$f(t, \phi) = (ae^t \cos(\phi),ae^t \sin(\phi), \int_0^t \sqrt{1-a^2 e^{2x}} dx)$$ I also calculated the matrix ...
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1answer
43 views

Elongation of edges of Moebius strip

Find length of edges of Moebius strip formed by cutting along length and re-joining from a circular cylinder segment: $( a \cos \theta, a \sin \theta, z), ( \theta, 0, 2 \pi ),( z,0,b) $ after half ...
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0answers
12 views

generalization of developable helicoid

Please indicate references to parametrization of a helicoid of constant negative Gauss curvature $ K = -1 $. Is it possible to modify the tangent developable helicoid parametrization in some way ...
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2answers
42 views

Second partial derivatives from first and mixed derivative of bicubic Bezier Surface Patch?

Given the definition of the bicubic Bezier Surface Patch function: $$f(u,v) = \begin{bmatrix} u^3 & u^2 & u & 1 \end{bmatrix} \begin{bmatrix} -1 & 3 & -3 & 1 \\ 3 & -6 ...
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1answer
33 views

Identify the Following Parametric Surfaces

1. $r(u,v) = ui+(u\cos v)j+(u\sin v)k$ 2. $r(u,v) = u\ cos(v)i+u\ sin(v)j+u^2k$ 3. $r(u,v) = ui+vj+(2u-3v)k$ 4. $r(u,v) = vi+\cos vj+\sin vk$ My Guess: Plane Circular Cylinder Cone ...
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1answer
27 views

what is the definition of “two parallel copies of a surface S”

As indicated in the title, suppose $S$ is a surface with genus $g$, then what is the definition of "two parallel copies of S"?
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1answer
168 views

Confusing Analysis proof

I have a question about a proof of the Beltrami-Enneper theorem: In the following $\nu$ is the surface-normal and $e_1,e_2,e_3$ the Frenet 3-frame. It states: Every asymptotic curve $c: I \rightarrow ...
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1answer
49 views

On a flat surface, can a holonomy can be nontrivial around certain curves

On a flat surface, can a holonomy can be nontrivial around certain curves? How is this possible?
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31 views

Compact hypersurface in $\mathbb{R}^n$

Let $S$ be an $(n-1)$ dimensional hypersurface in $\mathbb{R}^n$. If we say that $S$ is compact, does this necessarily mean that $S$ has no boundary? Eg. $S$ can be a sphere but not a sphere cut in ...
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1answer
36 views

Mutlivariable Calculus: Surface Area

This was a question a students had asked me earlier today regarding surface area. Find the surface area of the hemisphere $x^2+y^2+z^2 = 4$ bounded below by $z=1$. I decided to approach ...
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1answer
46 views

obtaining a surface equation by rotation

Let $C$ be a curve on the plane $(xoz)$ given by equation $g(x,z)=0$. How to find the equation in cylindric coordinates of the surface obtain rotating $C$ around zz axis?
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1answer
50 views

Blow-ups followed by contractions

Let $S$ be a minimal, non-singular complex projective surface. $\widehat S$ is the surface obtained by $r$ blow-ups of $S$ at the points $x_1,\ldots,x_r\in S$. Clearly $\widehat S$ contains exactly ...
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1answer
112 views

A “trivial” implication I don't understand.

I'm reading the article "Belyi's theorem for complex surfaces - Gabino Gonzalez Diez" and there are few lines of a certain proof that I don't understand (the author claims that all is trivial): ...
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1answer
50 views

Surfaces on which not every pair of points is connected by a geodesic

Let $S$ be a surface in $\mathbb{R}^3$. I believe that, if $S$ is smooth, bounded, and closed, then, for every pair of points $x,y \in S$, there is at least one geodesic $\gamma$ connecting $x$ to ...
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1answer
86 views

Proof of the existence of Lefschetz Pencils.

Let $S$ be a smooth complex projective surface. A Lefschetz pencil over $S$ is a rational map (which is not a morphism) $f:S--\rightarrow\mathbb P^1_{\mathbb C}$ with the following property: All but ...
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150 views

3D equation of a cone-like shape

Imagine there are two parallel planes (base plane and plane1) in the following image: There is one point on the base plane and there are several points on the plane1. The positions of these points ...
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35 views

Inverse function theorem and parametric surfaces as graphs

Let $\psi$ be a regular surface at the point $(u_{0}, v_{0})$ ($\psi \in C^{1}, T_{u} \times T_{v} \neq 0$ at $u_{0}, v_{0}$). Use the implicit function theorem to show the image of $\psi$ near ...
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3answers
65 views

surface area of the solid (column side)

I made a problem But I'm stuck in solving .. :-( the problem is following. Find the surface area of the solid that lies under the paraboloid $z =x^2 + y^2$, above the $xy$-plane, ...
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0answers
32 views

Torsion of an asymptotic curve with nonzero curvature

I wish to solve the following problem using the matrix of the shape operator $S_{P}$. Suppose $K= \text{det} S_{P} <0$, and $C$ is an asymptotic curve with curvature $\kappa (P)$ nonzero. I want to ...
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1answer
61 views

minimum number of points on the surface of a 3D ellipsoid to define it uniquely

An ellipsoid in 3 D is described by 9 independent parameters: 3 for the coordinates of its centre + 6 independent components of a symmetric 3 x 3 matrix. What is the minimum number of points on the ...
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2answers
63 views

Parameterizing a surface

The question I was asked goes like this: The part of the hyperboloid $5x^2 − 5y^2 − z^2 = 5$ that lies in front of the yz-plane. Let x, y, and z be in terms of u and/or v. Find a parametric ...
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0answers
32 views

Integrate Gaussian over elliptic area

I have got a two-dimensional gaussian distribution, where $\sigma_x = \sigma_y$ and $\mu_x = \mu_y = 0$. $ f(x,y) = \frac{1}{2\pi \sigma^2} e ^{-\left( \frac{x^2 + y^2}{2\sigma^2} \right)}$ I would ...
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1answer
28 views

Determine the best cup of coffee to have faster cooling possible

Assume that a cup of coffee is a cylinder. The coffee machine at my workplace always produces the same amount of coffee, so the volume is constant. The coffee is always really hot, so I'm looking (out ...
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1answer
61 views

Surface integrals and surface areas of arbitrary parameter domains

I'm having trouble evaluating this surface integral. This would be very simple to solve if the parameter domain of the variables u and u was a square region. However, that isn't the case here. I've ...
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2answers
94 views

Intersection and Curvature of Surfaces

a) Describe the intersection $(C)$ of sphere $x^2 + y^2 + z^2 = 1$ and the elliptic cylinder $x^2 + 2z^2 = 1$, and find out the total arc-length of this intersection. b) Determine the points on the ...
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37 views

Embedding Klein Bottle in $\mathbb{R}^4$ using a figure 8 loop.

I'm trying to show that we can embed the Klein bottle in $\mathbb{R}^4$. I've previously shown that a figure 8 curve can be embedded in $\mathbb{R}^3$ by a bump function that pushes away the ...
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2answers
614 views

Is the Euler characteristc defined wrong? If not, why not?

Ever since learning that $$\chi(S_0\# S_1) = \chi(S_0)+\chi(S_1)-2$$ (where $\chi$ denote the Euler characteristic), I've wondered whether $\chi$ isn't "defined wrong." If we let $\chi' = 2-\chi,$ ...
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2answers
88 views

Maps between Riemann surfaces are open and continuous

I'm having some trouble with a couple of concepts in Riemman surfaces that I would really appreciate some help clarifying! Firstly, is it true that a holomorphic map between two Riemann surfaces $f:R ...
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2answers
149 views

Function for a sphere

I believe that there is something fundamentally wrong with my understanding of functions but I can't pin point what it is, so I would greatly appreciate any guidance. Consider a unit sphere, ...
2
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1answer
101 views

Surface has Euler characteristic 2 iff equal to sphere

Let $\Sigma$ be a connected (not necessarily compact) surface with or without boundary. Is it true that $\Sigma$ is homeomorphic to the sphere if it has euler characteristic $\chi(\Sigma)\geq 2$? I ...
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1answer
98 views

Differential geometry: Conformal map

Let $f:\mathbb{R}_{>0} \times (0,2\pi) \rightarrow \mathbb{R}^3$ $$f(t,\xi) := (r(t) \cos( \xi) , r(t) \sin(\xi),z(t))$$ be a surface of revolution, where we assume that $r>0$ and ...
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2answers
99 views

Understanding how connected sum of smooth surfaces is a surface

I have two smooth surfaces $M_1$ and $M_2$ I''m trying to understand how the connected sum $M_1 \mathop{\#} M_2$ is a smooth surface. I will write my understanding of the proof and then explain where ...
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1answer
29 views

Understanding why Euler's Formula applies to planar graphs

I'm trying to prove that given a planar graph (by that I mean a graph where every pair of points is joined without crossings) $V-E+F = 2$. I can prove this by induction directly on the edges except ...
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2answers
51 views

Reference request for equality of torsion of H1 and H2

I have heard that for a surface $X$ (algebraic? smooth? compact?) the torsion part of $H_1(X,\mathbb{Z})$ is the same as that of $H_2(X,\mathbb{Z})$. Please could you give me a correct statement? I ...
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0answers
20 views

Cutting a surface at critical levels produces cylinders

Let $F$ a closed surface with isolated critical points and a homeomorphism $g: F \rightarrow F$ that maps critical levels to critical levels. Let us cut the surface $F$ by the critical levels. Do we ...
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3answers
343 views

Find the intersection of two surfaces

I have been looking into this question : we have two surfaces : $$\big\{(x,y,z)\in \mathbb{R}^3 \mid\;\; S_1\colon\;\; x+z=1 ,\;\; S_2\colon\;\; x^2+y^2=1 \big\}$$ we need to draw or describe the ...
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2answers
111 views

Quick question: Chern classes of Sym, Wedge, Hom, and Tensor

Given $L$ is a line bundle and $V$ is bundle of rank $r$ on a surface (compact complex manifold of dim 2). Recall the formula for $c_1$ and $c_2$: $c_1(V\otimes L)=c_1(V)+rc_1(L)$ ...
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0answers
52 views

Is there a surface S$\subset R^3$ whose Gaussian curvature is -1 at each point S?

Is there a surface $S\subset \Bbb R^3$ whose Gaussian curvature is $-1$ at each point $S$? At first I think this does not make a sense. But googling and googling.. I found a 'final exam problem' ...