For questions about surfaces.

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Cut-number of Klein bottle and other non-orientable surfaces

What is the maximum number $c$ (cut-number) of non-intersecting (edit: two-sided) circles on a Klein bottle $N_2$ and, in general, a surface $N_h$ with $h$ Möbius strips, such that cutting by these ...
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6 views

Parametrization of folded surfaces with symmetry

How is parametrization done for folded surfaces which are smooth within restricted interval of fold, e.g., has is it been possible to define parametrization for plane faces of Platonic solid faces? ...
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1answer
38 views

Christoffel symbols of a surface of revolution

I am looking for a way to write down the Christoffel symbols for a surface of revolution. They are given by ...
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74 views
+50

Squeezed cylinder parametrization

A Cylinder is such a common surface. But is there a parametrization for an isometrically $ R^2 $ bent cylinder whose major and minor dimensions are along x, y axes? I used an approximation to ...
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0answers
11 views

how can we make helical wave fronts with double twist [on hold]

How can we make helical wavefront with 2,3,4... twist? For an example of what I'm talking about, see the picture below:
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0answers
17 views

Calculating the equation of a multivariable surface of revolution

I'm stucked with a surface equation problem so I would be very thankful if someone could help me with it. What the excercise says: Find the equation of the revolution surface that is spanned when ...
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25 views

If Möbius band embeds then $\mathbb RP^2$ is a (connected) summand!

This is exercise 6-4 on page 181 in John Lee's Topological Manifolds book which asks me to prove the above, that is, if $M$ is a boundaryless surface which contains a subset $B$ which is homeomorphic ...
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27 views

An unclear passage in a proof of a theorem about complex surfaces.

Look at the following theorem (I have posted only part of the proof): I don't understand the highlighted sentence. Why the morphism $\widetilde{p}:\widetilde X\longrightarrow C$ can't map a ...
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1answer
31 views

What happens to geodesic curvature under the Gauss map?

$\def\RR{\mathbb{R}}$Let $D$ be a closed disc, smoothly embedded in $\RR^3$. The Gauss-Bonnet theorem tells me that $\int \!\! \int_D K + \int_{\partial D} \kappa = 2 \pi$, where $K$ is the Gaussian ...
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31 views

Meaning of “local equation” of a divisor.

Let $X$ be a smooth surface and moreover let $C,D$ be two effective divisors of $X$. Hartshorne says (page 357) that $C$ and $D$ meet transversally if the local equations $f,g$ of $C,D$ at $P$ ...
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2answers
33 views

Moebius band and Viviani Frill

I find a common rule that unites generation of Viviani Frill and the the Moebius Band. $$ \phi =\theta $$ where $ \phi,\theta $ are spherical coordinates. Please comment if this way looking at it ...
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1answer
38 views

The equation of a 3D surface bounded by 3 known elliptical curves

I am trying to find the equation of a 3D surface as illustrated below. The boundaries of this surface is comprised of two planar elliptical arcs $AB$ and $AC$ as well as a 3D arc $BC$ which is a 3D ...
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2answers
25 views

question about closed disc and closed surfaces.

Question: is a closed disc is a example of closed surface. I know that, the boundary of an open disk viewed as a manifold is empty, while its boundary in the sense of topological space is the circle ...
4
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2answers
52 views

Calculating Triple Integral

I have task : find volume of body limited by surface $(\frac{x}{a})^{2/3} + (\frac{y}{b})^{2/3} + (\frac{z}{c})^{2/3}$ = 1. I know that this task is about triple integral. But i have confused by such ...
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37 views

What are some elementary books which discuss projective lines on surfaces with examples?

I have the books: W. H. Blythe, On models of cubic surfaces (1905) and A. Henderson, The twenty-seven lines upon the cubic surface, and a couple more modern algebraic geometry books including I. R. ...
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0answers
34 views

What are some applications of parametrization of curves and surfaces?

I know that we can find all elements of a quadratic field with norm 1 by rational parametrization of conics, it can be used to show that some Diophantine equations are not so easy to solve, and that ...
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0answers
22 views

Non-standard 3D rotation of a set of points [duplicate]

I want to create a 3D surface as shown in the figure below. Toward this, I thought if I rotate a set of points in $xy$-plane on a elliptical arc I may be able to get such a surface. I was thinking of ...
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1answer
160 views

How to solve this questions about regular surfaces?

I'm trying to solve the following: $i)$ Show that if all normals to a connected surface pass trough a fixed point, the surface is contained in a sphere. $ii)$ Prove that if a regular surface $S$ ...
2
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1answer
63 views

Creating an ellipsoidal 3D surface

I am trying to find the equation of a 3D ellipsoidal surface. I have thought of two approaches which are schematically shown below: By revolving an elliptical arc over a 3D elliptical path: Or by ...
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1answer
40 views

Surface area of a 2-sphere in Abstract Index Notation

I believe the following completely specify a 2-sphere of radius 1 in AIN: $$ R_{ijkl}=\epsilon_{ij}\epsilon_{kl} \\ R_{ij}=g_{ij}\\ R_{ii}=g_{ii}=2 $$ It is easy enough to determine the area by ...
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1answer
35 views

Surface fitting to a mesh grid of data points

I wonder if there is a technique for fitting a surface to a given mesh grid of data points? I've seen interpolating a polynomial to $2$D data, but not $3$D. E.g. say I was given the matrix $$ ...
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1answer
42 views

Geodesic equation

Assume that you have a parametrization of a surface $f:\Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3,(u,v) \mapsto f(u,v)$. Now if I have a curve defined by $g(t)=f(0,t)$. The geodesic ...
3
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2answers
55 views

Differential Geometry of Curves and Surfaces

I'm self-studying differential geometry using Lee's Intro to Smooth Manifold and Do Carmo's Riemannian Geometry. However, I've never studied the subject so-called "differential geometry of curves and ...
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1answer
18 views

Evaluate Surface Integral over this triangular surface

When I solving the practice excericse problems at the end of the section, I stumbled upon this problem, which I have been trying to figure out how to compute the integral,but cant. Can someone please ...
2
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1answer
20 views

Is this cylinder a regular surface?

Let $C$ be a figure $‘‘8"$ in the $xy$ plane and let $S$ be the cylinder surface over $C$; that is, $$S=\{(x,y,z)\in\mathbb{R^3}:(x,y) \in C \}$$ Is the set $S$ a regular surface? I know that the ...
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1answer
41 views

Integrate the gaussian curvature

Let $T$ be a torus. We have a parameterization by $((c+a \cdot cos(v))cos(u),(c+a\cdot cos(v),a\cdot sin(v))$ for $u,v \in [0,2\pi)$. The first fundamental form is given by $E=(c+a\cdot cos(v))^{2}, ...
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1answer
27 views

Computing the first fundamental form

Let $X$ be a smooth surface in $\mathbb{R}^{3}$. I want to compute the first fundamental form of $X$. Assume that $X$ has 2 different local parameterizations $r_1$, $r_2$ (i.e. for $r_1$ there is a ...
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1answer
36 views

Does this surface exist

Does anybody of you know if there is a surface with first fundamental form $(g_{ij}) = \operatorname{diag}(1, \cos^2(u))$ and second fundamental form $(h_{ij}) = \operatorname{diag}(1, \sin^2(u))$? ...
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0answers
25 views

Equation for flat circular/lenticular surface

We know that $x^2+y^2= r^2$ is equation of circle curve but I want to draw a flat circular surface....not curve. i.e. to explain the problem-lets draw a flat circle not move it out of screen (positive ...
0
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1answer
24 views

Curvature line parametrization

I have a question about the curvature line parametrization. We said that for a given surface $f: U \rightarrow \mathbb{R}^3$ we find a local curvature line parametrization such that both the first ...
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0answers
49 views

Criterion for orientability: Derivative of transition map

The definition I have been given for a smooth abstract surface, $S$, to be orientable is that given a continuous family of maps $f_t: D \to S$ that embed the closed unit disk into $S$ with $f_0(D) = ...
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0answers
19 views

Delaunay surfaces - plane as surface of revolution

According to Wikipedia (and other sources) "Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These ...
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1answer
27 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
27 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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0answers
28 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 ...
2
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1answer
52 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
29 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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0answers
13 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
29 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
54 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
47 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
30 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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0answers
40 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
23 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
46 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
40 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
11 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
36 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
27 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
26 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"?