For questions about surfaces.

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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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23 views

Show that in these coordinates M is locally the graph $z=f(x,y) = \frac 12(k_1x^2 + k_2y^2) + e(x,y)$

Let us say that P is the origin and TpM is the tangent plane that is the xy-plane. We will let the x,y axes be the principal directions at P. Also, we will let the limit $$\lim_{(x,y)\to ...
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3answers
321 views

Surface area of a sphere limits

If I am finding the surface area of a sphere in spherical coordinates my intergral would be like this: $$\int^{\pi}_0 \int^{2\pi}_0 R^2 \sin (\theta) d\phi d\theta =4\pi R^2$$ But if I do the ...
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1answer
25 views

Analytically isomorphic fibers.

Suppose that $S$ is a non-singular complex projective surface with a fibration $f$ over $\mathbb P^1(\mathbb C)$. Suppose also that: There are only finitely many points $y_1,\ldots,y_n\in\mathbb ...
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2answers
28 views

Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured ...
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1answer
34 views

a problem with Stokes' theorem(curl)

If L is the circle which you get from the intersection between the sphere $$ x^2+y^2+z^2=1, y=x\sqrt(3) $$ and $$ I= \int_L (y-z)dx+(z-x)dy+(x-y)dz $$ so |I| equals to? but i dont understand how the ...
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20 views

Formation of PDE [closed]

I need to find the PDE arising from the surface: $f(x^2-y^2, x^2-y^2) = 0$ How can I approach such problems when surface is given as an arbitrary function of two complex functions ?
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29 views

Roulette of a parabola - Delaunay-Surface

I've problems to understand an equation I've found in various books and papers. Maybe someone could help me and explain it a little bit more precisely. I colored the equation in yellow (the picture ...
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20 views

Given the equation for a surface, how to find enclosed volume?

Suppose we give an equation of the form $f(x_1,x_2,..., x_n)=C$, with $f$ a smooth function, and assume this is such that defines a closed surface in $\mathbb{R}^{n+1}$. Assume also that the equation ...
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0answers
17 views

Principle Lines of curvature

In the text by Manfredo P. Do Carmo entitled Differential Geometry of Curves and Surfaces, an analysis of the principle directions is made near a non umbilic point on pp 160-161. I have followed his ...
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0answers
39 views

The curvature of surfaces in Euclidean space (Theorema Egregium)

The below animation is from Wikipedia. It shows how a helicoid can be deformed into a catanoid and vice versa without stretching. Because of this, the Theorema Egregium shows that the Gaussian ...
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1answer
33 views

Equation of ellipsoid surface obtained by revolving an ellipse

I'm working through the following example from the Princeton Review book: If the ellipse $x^{2} + x^{2/9}=1$ in the $xz-$plane is revolved around the $z-$axis, what's the equation of the resulting ...
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1answer
19 views

Asymptotic Directions of a Cylinder

Say I am looking at a cylinder. I have found the shape operator and I have found the eigenvalues to be k1 = -1/a and k2=0. I have also found the principal directions {1,0} and {0,1}. I know that if ...
3
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2answers
46 views

Differential Geometry: Is a closed disk a surface?

An open disk is clearly a surface, in the sense that it is locally homeomorphic to a part of $\mathbb{R}^2$. But what about a closed disk, even though it still looks like a surface, I am starting to ...
2
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1answer
29 views

Proof that a map from an orientable surface to a non-orientable surface has even degree.

For a smooth map $f:M\to N$ from an orientable closed surface $M$ to a non-orientable closed surface $N$, we define its parity (also called modulo 2 degree, and denoted $\deg_2(f)$) as the parity of ...
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1answer
29 views

Calculus 3 - Level surfaces

So I know how to find the level surfaces for a $2$ variable functions, $z=(x,y)$, by finding the $3$ planes. How would you find the level surfaces for a $3$ variable function, $w=(x,y,z)$. Would you ...
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1answer
34 views

slope of a curve in $\mathbb{R}^3$

The surface given by $z = x^2 -y^2$ is cut by the plane given by $y = 3x$, producing a curve in the plane. Find the slope of this curve at the point $(1, 3, -8)$. My answer is: $$f(x, y, z) = x^2 - ...
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1answer
24 views

Vector Parametrization of a Hyperbolic Paraboloid and a Plane

So I need to find the intersection between a hyperboloid ($z=\frac {y^2}{b^2}-\frac{x^2}{a^2}$) and some related plane ($bx+ay-z=0$). I have tried solving for $z$ and equating the two: ...
2
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1answer
22 views

Surface area of transformed sphere

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface ...
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0answers
53 views

How to improve my proof that a parameterized surface x(u,v) is conformal if and only if E=G and F=0

I am trying to prove that a parameterized surface x(u,v) is conformal if and only if E=G and F=0 . So far I know the following, I understand that I should use the Gaussian first fundamental form of ...
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1answer
36 views

Ruled Surface: $x(u,v) = \alpha(u) + v\beta(u)$, one of the following is true

Say we have a ruled surface that is given by $x(u,v) = \alpha(u) + v\beta(u)$ with $\alpha'$ not equal to $0$ and $\| β \| = 1$. If $\alpha'(u), \beta(u)$, and $\beta'(u)$ are linearly dependent for ...
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0answers
46 views

geodesic polar coordinate parallel circles

When is it possible to have the same constant geodesic curvature on all parallels of a constant Gauss curvature surface? EDIT: picture added.
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Revolution surfaces of constant Gaussian curvature k=1

Prove that all revolution surfaces $(\phi(v) \cos u ,\phi(v) \sin u,\psi(v)) $ of constant Gaussian curvature $k = 1$ is one of the following types: $\phi(v)=C\cosh v$ and $\psi(v)=\int_0^v ...
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1answer
40 views

How to calculate volume and surface area of three dimensional figures given set of three dimensional coordinates?

I have set of three dimensional coordinates, and the shape is unknown. I would like to calculate the surface area and volume for these coordinates approximately. What is the right approach to solve ...
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1answer
36 views

Finding the surface area of a parametrized surface

I was wondering how you would compute the surface area of a parameterized surface. Is there a formula or set of procedures you can follow to compute this. Say I wanted to compute the surface area of a ...
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0answers
24 views

Orientation of surfaces

From the book: Fixing a parametrization $x(u,v)$ of a neighborhood of a point $p$ of a regular surface $S$, we determine an orientation of the tangent plane $T_p (S)$, namely, the orientation of the ...
3
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1answer
48 views

Inverse mapping for a simple $\mathbb{R}^3$ surface given by $(\sin u, \sin 2u, v)$.

For a domain $U=\{\, (u,v) \in \mathbb{R}^2 \mid -\pi<u<\pi,\ 0<v<1 \,\}$ we have a mapping $X \colon U \to \mathbb{R}^3$ defined by $X(u,v) = (\sin u, \sin 2u, v)$. The resulting surface ...
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0answers
105 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
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26 views

Two unit disks spliced together?

$$ x^2 + (y-z)^2 = 2 x^2 z/y $$ The surface represented by above equation is formed by radial cuts on two separate unit diameter disks spliced together forming a "continuous" surface around ...
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27 views

“Circle” on pseudosphere

How should parametrization of the 2 parameter surface of a pseudosphere ("latitude" u and longitude v) change to result in a 1 parameter curve of constant geodesic curvature? EDIT: In other ...
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2answers
45 views

Viviani on Sphere parametrization

How should parametrization of the 2 parameter surface of a sphere (latitude u, longitude v) be changed to result in 1 parameter curve of Viviani?
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1answer
67 views

The projections are differentiable $1$-forms.

Suppose $M$ is a surface and suppose $X: U \subset \mathbb R^2\rightarrow M$ is a coordinate patch. Then for every $p \in X(U)$, the pair of vectors $(X_u(X^{-1}(p),X_v(X^{-1}(p) )$ is a basis of the ...
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1answer
55 views

Curvature proof of a convex plane curve

Having a little trouble with part b. Is there a way to show that this curve would be arc length paramaterized? I am assuming that we cannot say this. If it is not we can take alpha', alpha'' and ...
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1answer
60 views

Give a closed plane curve C with k (curvature) > 0 that is not convex, Draw closed plane curves with rotation indices 0, 2, -2, and 3

1.) Give a closed plane curve C with k (curvature) > 0 that is not convex. can someone please explain these concepts to me. How can you have a closed plane curve like this? Do you used signed ...
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0answers
26 views

sphere parametrization

We have the standard spherical surface parametrization in which one set describes geodesics (longitudes) and another ( latitudes/parallels). What parametrization may be possible for both sets to be ...
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0answers
37 views

A converse proof that involves Torsion, curvature, and differentiation that equates to 0

I am having difficulty proving the converse in part B. I understand part A and have shown that t/k-t/k = 0. I found that n=-1/k, n'-bt= b'+tn = 0, so n' = bt and b'=-tn. However, I am unable to find ...
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2answers
90 views

A curve internally tangent to a sphere of radius $R$ has curvature at least $1/R$ at the point of tangency

Suppose $a$ is an arc length-parametrized space curve with the property that $\|a(s)\| \leq \|a(s_0)\| = R$ for all $s$ sufficiently close to $s_0$. Prove that $k(s_0) \geq 1/R$. So, I was going ...
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2answers
42 views

surface presentation

Given the following group and presentation, how could I go about showing if there exists a compact surface with that fundamental group? The group is $\big \langle a, b, c, d, e $ $\mid$ ...
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1answer
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Total variation as surface area smooth functions of two variables.

I learnt we have different definitions for the total variation for functions of the form $f:\mathbb{R}^2\to\mathbb{R}$ which are in some way analogous to the total variation of functions of one ...
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0answers
43 views

What does this operator $\odot$ mean

I read this about the second fundamental form in Wikipedia and I’ve no idea what does $\odot$ mean? Does anybody know? $$II=-dN\cdot dP=\omega^3_1\odot\omega^1+\omega^3_2\odot\omega^2$$
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1answer
13 views

Equation for multiple peak surface to test particle swarm optimising algorithm

i have developed a particle swarm optimisation algorithm that i am running some tests on. It is able to solve simple equations like this: $x^2 + y^2 + 300y - 254x + 3$ with only one optimum but ...
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3answers
53 views

Proof that this surface is of revolution

I have a surface with parametric equation $$\mathbf{x}(u,v)=(u\cos(v),u\sin(v),u^2),$$ $u$ is any real number, $v$ is between $0$ and $2\pi$. I don't know how to show that this is surface of ...
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0answers
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lifting loops on surface with abelian fundamental group for decrease their self-intersection number

I found a proof of following statement (which is available here ), but I'm not sure if we must assume that fundamental group of surface $M$ is non-abelian: Lemma: Let $M$ be a compact orientable ...
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1answer
36 views

Classical presentation of fundamental group of surface with boundary

It is well known fact about fundamental group of orientable compact surface: Letting $g$ be the genus and $b$ the number of boundary components of surface $M$. There is a generating set ...
2
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1answer
220 views

Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
2
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1answer
22 views

Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
2
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0answers
45 views

Finding the leftmost, rightmost, top, and bottom, points, on a surface, of a sphere.

So I'm making a 3D game, and the player is inside a glass sphere. I'm projecting a bunch of points onto the sphere, and I need to find the leftmost, rightmost, topmost, and bottommost points, so I can ...
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2answers
51 views

1-dimensional foliation on a surface

Is it possible to find a 1-dimensional nonsingular foliation on an orientable surface with one boundary component such that lines of the foliation are transverse to the boundary?
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1answer
42 views

Why does every noncompact orientable surface have a complex structure?

There is a high-powered proof of the fact that orientable noncompact surfaces have free fundamental group here that invokes the ability to put a complex structure on any such surface. But why should ...
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1answer
19 views

How do I find the ridges and valleys given a surface elevation function

Given a surface with a single elevation value for every x and y how can I find the places where the isoelevation contours have the tightest bends? And how can I differentiate between bends that are ...