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

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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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38 views

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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19 views

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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69 views

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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67 views

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 $mK_S$ ...
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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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32 views

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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85 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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54 views

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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32 views

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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1answer
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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67 views

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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50 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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51 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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56 views

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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116 views

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 ...
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148 views

Surface area common to two perpendicularly intersecting cylinders

I need help to calculate the following surface area: the surface area common to the two cylinders $x^2 + y^2 = a^2$ and $x^2 + z^2 = a^2$ using surface integrals essentially. My attempt: Let ...
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157 views

Deriving the surface area of a sphere from the volume

I am a high school student, so I know how to derive the volume $V=\dfrac{4}{3}\pi r^3$ using calculus, but I am unable to derive its surface area. However, I notice that we can approximate the ...
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27 views

Verification of a proof regarding the connected sum of two surfaces

I am trying to solve the following exercise: Let $X_1, X_2$ be two surfaces. Lets consider charts $\varphi_j: U_j \to \mathbb{R}^2$ with $U_j \subset X_j$, $j= 1, 2$ and let $B_j = ...
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What does it mean to 'preserve the first fundamental form'?

I'm a bit confused about the phrase 'preserving the first fundamental form', or 'The Gaussian curvature is determined by the first fundamental form'. For example, let's say I have two surfaces $M$ ...
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Intersection of curves on surfaces and the sheaf $\mathcal O_{C\cap C'}$

Let $S$ be a (complex) projective surface and let $C,C'\subset S$ two closed irreducible curves (namely two prime Weil divisors). It is well defined the scheme $C\cap C'$ (as fibered product of ...
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How to classify this surface

I know that it should be either a sphere, torus, Klein bottle, real projective plane, or a connect sum of any combination of these, but I don't know the steps in identifying what kind of surface ...
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86 views

What's the name of these two surfaces?

I've plot two implicit surfaces which are shown in the above, I only know their expression, but I don't know how to call them.
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52 views

Effective Weil divisors on a Surface and local equations

Let $S$ be a non singular surface over an algebraically closed field $k$ ($S$ is a $k$-scheme integral, of finite type and separated). Suppose that $D\subset S$ is an effective Weil divisor; I don't ...
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35 views

Identify the direction of a loop of Edges on a Face

I have a geometrical face which is defined by a surface. The face is also defined by a few edges which are curves, they bound the face. These edges have start and end points. They flow in a loop ...
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74 views

Why $\sigma(u,v)=(\cos u\ \cos v,\cos u\ \sin v,u)$ is not a surface?

During evaluation of different special cases of Gaussian curvature for the surface of revolution in Ch. 7 (page 152) of Elementary Differential Geometry by Pressley, it comes to the case when ...
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52 views

Showing that the closed unit disk is not a regular surface

First of all, I realize that this question has been asked on this site before; but looking at the answers already given none of them go into the details which are causing me trouble (all the relative ...
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39 views

Is the given set an open subset of the $\mathcal{G}^r_d(|L|)$

Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be a very ample line bundle on $X$. We have a variety $\mathcal{G}^r_d(|L|_s)$ associated to the linear system of curves $|L|$. The ...
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97 views

Existence of a vector field with one singularity on a surface

In a paper I'm reading, it is stated that "It is known that on a compact, connected, oriented two dimensional manifold, there exists a vector field with only one singularity". Where can I find a ...
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23 views

Showing that the intersection of two particular vector spaces has codimension 1 in the smaller of the two spaces

Here is the setting: I have a compact symplectic manifold $(X^{2n},\omega)$ and a compact symplectically embedded submanifold $(M^{2d},\sigma)$; that is, $\iota^{\ast}\omega=\sigma$. The dimension 2d ...
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50 views

find resultant XYZ points in a laser 3D scanner

I am working on a 3D scanner using a laser and camera and some other stuff needed, the idea is that the camera captures an image to the target when the laser is on, then use the image to gain the ...
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The Walking Saddle Curve and other weirdly moving shapes

The link at the end of this sentence seems to be a video of a walking saddle curve. Where can we obtain these? What are more exact parameters of this curve? What is the relation between wind speed ...
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Measure on surfaces in $\mathbb{R}^3$

I am interested in the following three surfaces in $\mathbb{R}^3$: $$S_1=\{(x,y,x+y+x^3+\sqrt{y}): x \in [0,x_0], y\in [0,y_0]\},$$ $$S_2=\{(x,y,x+y+x^2 +y^2+1): x \in [0,x_0], y\in [0,y_0]\},$$ ...
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Developable surfaces in $\mathbb{R}^4$

It is known that there are developable surfaces in $\mathbb{R}^4$ which are not ruled: the famous example is of Hilbert and Cohn-Vossen in their book "Geometry and the Imagination" (p. 342). The ...
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Response Surface Methodology using Moving Least Squares Method

I would like to obtain the response surface of a mathematical function for reliability-based design optimization (RBDO). To obtain a reliably response surface, I learned that moving least squares ...
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91 views

converse of theorema egregium

Suppose $(M,g)$ is a $3$ dimensional Riemannian manifold and $N$ is any surface imbedded in $M$. If the theorema egregium holds for $N$ does it follow that $M$ is flat? The way I'm thinking of the ...
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28 views

Principal directions of surface given by data points

I am wondering how I can easily find the principal directions of a surface given by data points. The points are given in a fixed matrix. Assuming that the surface is differentiable, I can easily find ...
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Parametrization of helicoid like surface for Faraday's law of induction of a solenoid?

I want to visualize with mayavi a possible surface for Faraday's law of induction in the electrodynamics of a solenoid. I.e. something like a helicoid with a smooth transition to a rectangular area, ...
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107 views

Grothendieck and Segre's proof of the Hodge Index theorem

On a field of positive characteristic we have the following well-known and important result: (Hodge Index theorem) Let $H$ be an ample divisor on a surface $X$, and suppose that $D$ is a ...
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What is the mean curvature of a developable surface?

I am trying to calculate the mean curvature of a developable surface but having some difficulty. Any help would be greatly appreciated! I am very new to differential geometry so please bear with me, ...
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44 views

Surface with all tangent plane pass through one common point is a cone surface

I want to prove that a surface with all tangent plane pass through one common point is a cone surface (a surface like this $\mathbf{r}(u,v)=\mathbf{a}_0+v\mathbf{b}(u)$, here $\mathbf{a}_0$ is a ...
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Why $\dfrac{\partial \sigma}{\partial u}=\dfrac{\partial \sigma}{\partial \bar{u}}$?

According to Elementary Differential Geometry by A N Pressley: $\Large\textbf{5.3. Conformal Mappings of Surfaces}$ Now that we understand how to measure lengths of curves on surfaces, it is ...
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45 views

Is there affine surface of general type of the form $y^2=f(x) f(z)$ or $y^2=f(x) g(z)$?

Let $f,g$ be univariate polynomials with integer coefficients of degree $n$. Is there affine surface of general type of the form (1) $y^2=f(x) f(z)$ or (2) $y^2=f(x) g(z)$? I would expect for $n$ ...
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1answer
40 views

Surface of $(x^2 + y^2 + z^2)^2 = a^2 * (x^2 - y^2)$ using surface integrals

I have to find the surface of $$(x^2 + y^2 + z^2)^2 = a^2(x^2 - y^2)$$ using a surface integral and really have no idea what to do... I would really appreciate it if you could give me an idea.
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83 views

Find the maximum volume of the pyramid bounded by the plane and the coordinate planes?

Surface $\sqrt{c}=\sqrt{x}+\sqrt{y}+\sqrt{z}$ , $(c>0)$ I found that at $(x_{0},y_{0},z_{0})$ a tangent plane to the surface is : ...
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83 views

Differential geometry: restriction of differentiable map to regular surface is differentiable

From Do Carmo: Let $S_1$, $S_2$ be regular surfaces. Suppose $S_1\subset V\subset \mathbb{R}^3$ and $\varphi:V\rightarrow \mathbb{R}^3$ is a differentiable map such that $\varphi(S_1)\subset S_2$. ...
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46 views

Computing Gauss curvature using Meusnier theorem

I have troubles with finding Gauss curvature and mean curvature in a certain point of an oblique cylindrical surface. I know the way using the fundamental forms, but I am supposed to use the Meusnier ...
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47 views

Why can't the pseudosphere be completed in $R^3$?

Without appealing to Hilbert's theorem on the non-embeddability of complete hyperbolic surfaces in $R^3$, is there a way to "see" that one can't extend the pseudosphere / surface of revolution of a ...