Surface is a two-dimensional submanifold of three-dimensional Euclidean space.

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Decomposing a surface $S$ with a simple closed curve $\Gamma$

In class we learned about how the Euler characteristic changes when we take a connected sum of surfaces $M_1$ and $M_2$: $$\chi(M_1 \# M_2) = \chi(M_1) + \chi(M_2) - 2,$$ and it made me wonder how ...
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40 views

What exactly is a surface integral?

I'm learning surface integrals right now and I don't think I fully understand what they are. What exactly do surface integrals represent? Is it volume? The basis for surface integrals seems just like ...
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50 views

How big are regular (hyperbolic) polygons?

Given a hyperbolic surface of constant curvature $K=-1/a^2$ embedded in $\mathbb{R}^3$, is there a known formula for the length of the edges of a regular polygon? I know that the Gauss–Bonnet ...
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23 views

Extend simplicial homeomorphism in a PL surface

Let $S$ be a connected PL closed surface. How can I show that, given a 2-simplex $\Delta$ in $S$ and a simplicial homeomorphism $g:\Delta\to \Delta$ that preserves orientation, this can be extended to ...
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27 views

Prove a surface is a plane

Let $\Sigma$ be a $C''$ surface defined on an open connected set D in the UV plane. Suppose $d^2\Sigma=0$ in $D$,prove that $\Sigma$ is a plane. I know $\Sigma$ has the form ...
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45 views

How to find contact points of a plane with an uneven surface

I try to find the contact points of a plane when it is placed on an uneven surface. For example a book that is placed on uneven terrain, where would it touch the ground? I already have some ideas how ...
3
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38 views

Images of linearly equivalent divisors are linear equivalent?

Let $f:X\longrightarrow Y$ be a finite morphism of degree $d$ projective surfaces over $\mathbb{C}$. Suppose that $X$ is smooth. Let $L'$ be a line bundle on $Y$, and let $L=f^*L'$. 1) Consider a ...
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How to convert density function to hermite data?

[Apologies for any mistakes in tagging, I'm new to this site and topic] I'm rendering a procedural surface in a game engine. A method returns density for a particular point in space. Density is a ...
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56 views

How to parametrize the surface $x^3 + 3xy + z^2 = 2$ and compute a tangent plane

How do I parametrize the surface $x^3 + 3xy + z^2 = 2$ and compute the tangent plane at $(1, \frac{1}{3}, 0)$ using the resulting parametrization? I know that the tangent plane should be ...
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45 views

Misinterpretations of Hilbert's Theorem?

I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance: "I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ ...
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180 views

Fundamental forms of constant mean curvature surfaces

For surfaces of constant mean curvature in $E^3$, prove that either they are all-umbilic-points surfaces or their fundamental forms can be represented as following: I $=\lambda(u,v)(dudu+dvdv)$ ...
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Isomorphism on Homology induced by Inclusion of CW-complexes

I want to prove the following and I do not know how. Let $X$ be a CW-complex of dimension $n-1$ and let $Y$ denote the space obtained from $X$ by attaching a finite number of $n$-cells. Then the ...
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1answer
59 views

How to interpret a 3D plot

I am trying to interpret 3D plots but its driving me nuts. Lets say I want to see the relationship between 3 variables and I have data as following: ...
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58 views

Compute flux of a vector field with divergence of zero (multivariable calculus)

I'm attempting to solve this problem: Compute the flux of the vector field $\vec F = (e^z − 2xy, y^2 − e^z, 2xy − y^2)$ through the surface S that is the part of the surface $ z = x^4 + e^{y^2} $ ...
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32 views

Parametrization of a rotating surface

What is the parametrization of a surface obtained by rotating the circle $(y − 3)^2 + z^2 = 1, x = 0$ about the z-axis. I came up with the parametrization $S(r,θ) = (r , 3+cosθ , sinθ)$, is it ...
2
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47 views

Generally compute Gauss curvature

For a surface $F(x,y,z)=0$, we want to compute its Gauss curvature. I tried to suppose $z=f(x,y)$ locally and get a complicated expression. Is there any direct way to compute this? Thanks for your ...
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83 views

When are mapping tori isomorphic as bundles over the circle?

Suppose $\Sigma$ is an orientable genus-$g$ surface (possibly with boundary). The mapping torus corresponding to an orientation-preserving diffeomorphism $\phi: \Sigma \to \Sigma$ is the quotient ...
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Manifold is cut along hypersurfaces; how to define a connection on this?

May $M$ be a smooth manifold with boundary $\partial M$. Metrics and Connection can be defined everywhere. But now this manifold is cut by a smooth hypersurface $A$ and the cut goes along $M \cap A$ ...
3
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96 views

Non-existence of embedded incompressible surfaces

I want to prove the following assumption: Let $g,h$ be natural numbers with $g > h$ and let $S_g$ be the closed, orientable surface of genus $g$. Then, there is no (smooth) map $f: S_g \to S_h ...
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34 views

Path Lengths on a Sphere example.

What does $\rho$ mean as to be measured along geodesics and more importantly how would i be able to parametrize this accordingly as being on the sphere's surface? I know that I have to use the First ...
2
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1answer
137 views

Find the Equation of the Envelope of a Family of Line (Plane) Segments

Consider the first quadrant in the $OXY$ plane in $\mathbb{R}^2$. Point $O$ is the origin and the points $P$ and $Q$ are chosen on the $y$-axis and the $x$-axis, respectively as it is showed in the ...
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52 views

what is the minimum surface area shape required in order to contain a 1 meter line at all angles

been stuck on solving/proving the following puzzle: You need to make a hole in the wall, so that a 1 meter line can pass it through the hole at all angels, find a shape with minimum surface area that ...
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1answer
34 views

For all unit vectors $\mathbf u$ and a positive definite $\mathbf C$, what surface do vectors $\mathbf u \mathbf u^\top \mathbf C \mathbf u$ form?

Let $\mathbf C$ be a positive-definite $k\times k$ matrix. For all vectors $\mathbf u\in \mathbb R^k$ of length $\|\mathbf u\|=1$, consider vectors $\mathbf {uu}^\top\mathbf{Cu}$; they form a surface ...
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A surface with Euler characteristic of $-1$ [closed]

Is it possible to a have a surface that has an Euler Characteristic of $-1$ and what would that surface be homeomorphic to? $\displaystyle \chi \left({M}\right) = -1$
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scalar curvature under conformal deformation of a two - dimensional Riemannian manifold

I am currently stuck with an identity that I'd love to derive myself. Suppose $(M,g)$ is a surface (a two - dimensional Riemannian manifold) without boundary. Let $\tilde g = e^{2u} g$ be a conformal ...
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The Hirzebruch surface $\mathbb F_n$ contains a unique irreducible curve with negative self-intersection.

Consider the $n$-th Hirzebruch surface $\mathbb F_n:=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}\otimes\mathcal O_{\mathbb P^1} (n))$, and let $C_0\subseteq \mathbb F_n$ a section of the ruling ...
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1answer
55 views

$\mathbb P^1\times \mathbb P^1$ is minimal

I'm trying to prove that the complex surface $\mathbb P^1\times \mathbb P^1$ is minimal. I'd like to prove it directly, namely by showing that there are not exceptional curves. Suppose that $E$ is a ...
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34 views

Is equal curved surface areas a coincidence

If we take a cylinder of height $2x$ and radius $x$, as well as a sphere of radius $x$, we notice that they have the same curved surface area. Also, if we take the frustum of a cone such that it has ...
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96 views

Stokes's theorem outer unit normal for a hemisphere

Stokes's theorem says $$\oint \mathbf{F}.d\mathbf{r}=\iint_{D}^{-} \nabla \times \mathbf{F}\cdot \mathbf{n} \, dA$$ Evaluate using the RHS of Stokes' theorem the following problem. for the ...
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Geometrically ruled surface, sections and intersection numbers.

Consider a geometrically ruled surface $\pi: S\longrightarrow C$ where both $S$ and $C$ are projective, non-singular and complex ($C$ is obviously a curve). By using Tsen theorem one can show that ...
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Pullback of line bundles and divisors

Let $X$ and $Y$ be smooth projective surfaces over an alg. closed field. Let $f:X\longrightarrow Y$ be a finite morphism of degree 2. Let $C$ be a smooth curve which maps to a smooth $C'$ under $f$. ...
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Finding Gauss curvature of helicoid

For a helicoid with parametrization $$ ( u \cos v, u \sin v, f(v)) $$ it is required to find f(v) for a constant negative Gauss curvature $ -1/a^2. $ By going through procedure with Christoffel ...
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31 views

Why does $ds=\frac{dxdy}{|n\cdot k|}$ for surface integrals?

I have come across the answer to a surface integral here: http://image.slidesharecdn.com/presentation1-130305202701-phpapp01/95/integral-permukaan-15-638.jpg?cb=1362515311 And at one stage it says: ...
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Relation between curves in a complete linear system contained in another

Let $X$ be a projective surface over $\mathbb{C}$, let $x\in X$ be the only singular point of $X$. Let $L$ be an ample line bundle on $X$. Consider the blow up $Y$ of $X$ along $x$, ...
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How to rotate two of three axis of an object to the surface normal vector.

In this system every object has three axis: X, Y, and Z. I can cast rays at objects to get the information of the specific face the ray contacted. That information contains the surface normal of the ...
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About a theorem in Beauville's book.

Look at the following theorem (due to Castelnuovo) taken from Beauville's book on complex surfaces: I have a question about the behavior of $f$ on the exceptional curves generated by $\eta$. ...
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1answer
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Conformal immersions from surfaces into 3-manifolds

Let $f:(S,g) \to (M,h) $ be a smooth immersion of a compact surface into a 3 - manifold. Is it true that there exists a diffeomorphism $\phi: S \to S$, such that the metric $(f \circ \phi)^*(h)$ is ...
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if the canonical divisor is nef, then a multiple if effective

Let $S$ be a complex projective non-singular surface. Couldd you explain the following implication: If the canonical divisor $K_S$ is nef then there exists a number $m>>0$ such that $mK_S$ ...
3
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1answer
67 views

Second fundamental form and metrics

Suppose $M$ and $N$ are orientable manifolds, $f: M \to N$ is a smooth embedding and $g$ is a Riemannian metric on $N$. When $M$ has codimension $1$ and $\vec{n}$ is a prefered unit normal section of ...
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Is there a way to parametrise general quadrics?

A general quadric is a surface of the form: $$ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Eyz + 2Fxz + 2Gx + 2Hy + 2Iz + J = 0$$ It can be written as a matrix expression $$ [x, y, z, 1]\begin{bmatrix} A && ...
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1answer
84 views

What is the word associated with the connected sum of two surfaces with boundary?

I want to calculate the word associated with the connected sum of two surfaces with boundary but I don't know how to proceed. I know that the word associated with the connected sum of two surfaces ...
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1answer
75 views

Understand the corrispondence between rational maps and linear systems.

A key fact in "birational geometry" (on $\mathbb C$) is the following theorem: Let S be a surface. Then there is a bijection between the following sets: (i) {rational maps ...
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How to calculate Euler characteristic of surfaces $K$ and $P$?

The book Introduction to Topology by C. Adams and R. Franzosa says : From the triangulations in Figure 14.8, we see that $\chi(S^2) = 2$, $\chi(T^2) = 0$, $\chi(K) = 0$ and $\chi(P) = 1$. And ...
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RBF - triangle mesh interpolation - skinny triangles and incorrect results

I have triangle mesh, that I need to describe by RBF. I need to do this only locally on vertex neighborhood. All is working correctly if underlaying triangulation is reasonably regular. But if there ...
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36 views

Prove that $S$ is colorable if and only if it is orientable

I am taking a course on algebraic topology and I am trying to prove the following exercise: Let $S$ be a differentiable surface in $\mathbb{R}^3$. Prove that $S$ is colorable (you can paint one ...
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Can any surface be described by an equation?

There seems to be two definitions of a surface: The set $S$ of points $(x,y,z)$ satisfying the equation $f(x,y,z)=0$ for some smooth/differentiable function $f:E^{3} \to R$ with $\nabla f \neq 0$ on ...
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49 views

Is a point on a plane part of a face on the plane?

There is a line and a face in $\Bbb R^3$, does the line inersect the face? I have a plane (infinite area) in $\Bbb R^3$ defined by a point $(x_0,y_0,z_0)$ and its normal $n$. The plane contains a ...
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50 views

Relation between curves on a surface and divisors

Given a projective surface $S$, the irreducible curves contained in $S$ are exactly, by definition, the prime (Weil) divisors of $S$. I was wondering what are reducible curves on $S$ in terms of ...
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Is the principal curvature of a cylinder positive or negative according to the second fundamental form?

First off, what is the name of the tensor associated with the second fundamental form? For the first fundamental form, I believe we call the associated tensor, "the metric tensor." Principal ...
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Points on ellipsoid with maximum Gaussian curvature/mean curvature.

Find the points on the ellipsoid $$x^2/a^2+y^2/b^2+z^2/c^2=1$$ with maximum Gaussian curvature and mean curvature respectively. I parametrized it as $(a\sin u\cos v,b\sin u\sin v, c\cos v)$ and ...