For questions about surfaces.

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1answer
282 views

Numerical computation of surface curvature

In 2 dimensions, the definition of curvature of a curve $y = y(x)$ is \begin{equation} C = \frac{y''}{(1+y'^{2})^{3/2}} \end{equation} and it is easy to estimate the curvature numerically for given ...
4
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2answers
50 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 ...
-1
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0answers
20 views

What is connexity (in simple language)? [on hold]

Please explain the meaning of connexity, connex elements in simple language
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0answers
28 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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28 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
14 views

Non-standard 3D rotation of a set of points

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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0answers
76 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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0answers
37 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
39 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 ...
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2answers
51 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
102 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 ...
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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 ...
2
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1answer
47 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 ...
2
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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) = ...
3
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1answer
168 views

A 3-manifold with fundamental group isomorphic to a surface group.

Let $M$ be a 3-manifold (the case I am interested is $M$ closed orientable connected hyperbolic); suppose $\pi_1 (M)$ is isomorphic to the fundamental group of a (closed orientable connected) surface ...
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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 ...
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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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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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3answers
2k views

Geodesic of a curved surface

I'm trying to read Lambourne's Relativity, Gravitation and Cosmology, but as this seems more of a maths question I've posted it here rather than in the physics forum. The author talks about affinely ...
0
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1answer
353 views

Find the surface area obtained by rotating $y=1+3x^2$ from $x=0$ to $x=2$ about the y-axis.

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ...
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0answers
18 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 ...
2
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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 ...
5
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1answer
100 views

Number of fibrations over a curve.

Fix a non-singular complex projective curve $C$. I would like to know how many non-singular complex projective surfaces $S$ have the following properties (up to isomorphism): There is a fibration ...
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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 ...
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2answers
35 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
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 ...
0
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1answer
37 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
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
27 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 ...
7
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4answers
278 views

Why is $\pi r^2$ the surface of a circle

Why is $\pi r^2$ the surface of a circle? I have learned this formula ages ago and I'm just using it like most people do, but I don't think I truly understand how circles work until I understand why ...
0
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1answer
46 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 ...
0
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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 ...
1
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1answer
45 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 ...
2
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0answers
34 views

Parameterization of surfaces of revolution with constant mean curvature

Let $x(u,v) = (g(u), h(u) \cos v, h(u) \sin v)$ be a parameterization of a surface of revolution $M$, arising from rotating the regular curve $\alpha(u) = (g(u),h(u),0)$ around the $x$-axis with ...
0
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1answer
29 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} - ...
3
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1answer
262 views

If radial projection is bijective then is it a homeomorphism?

Suppose $S$ is a regular surface in $\mathbb{R}^3 $ and $0\not\in S$. Now consider the radial projection $f: S\to\mathbb{S}^2$ given by $$f(x)=\frac{x}{||x||} \hspace{5mm}\mbox{ for all $x\in S$}$$ ...
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2answers
143 views

Does a Möbius strip have only one shape? Or may it have different shapes?

I'm reading a book about geometry, and after thinking and viewing the Möbius strip, I want to know whether the book is right or not. The book says with a little description (that I can't write here ...
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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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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 ...
0
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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"?
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4answers
201 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
3
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1answer
141 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 ...