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

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Is there a pluricanonical divisor on a relatively minimal complex elliptic surface that can be written as sum of fibres?

A complex algebraic surface $S$ is said to be elliptic if there are a smooth curve $B$ and a surjective morphism $p \colon S \to B$ whose generic fibre is an elliptic curve (i.e. a smooth curve of ...
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28 views

Choose the reflection planes of a surface through a single point.

Let $S$ be a surface in $R^3$, for which coordinate vector field of $S$ has zero mean on $S$. Assume that for any vector $n$, a normal plane to $n$ exist, such that $S$ is symmetric about it. How can ...
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On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
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23 views

How to work out the angle of a line passing through a plane

I have a triangular plane composed of three points. From this it it easy to deduce that the plane is in fact composed of two vectors which must touch at some point. because all of this is relative, ...
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74 views

Does intrinsic mean existing regardless of some bigger space?

How is the arc-length of a regular parametrized curve in a surface $S\subset\mathbb{R}^3$ intrinsic? Let $\bf{x}\rm(u,v)$ be a parametrization of $S$. Letting $E,F,G$ denote the coefficients of the ...
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20 views

Question about determinig types of surfaces?

$$x^2 +y^2 +z^2 +2x +1=0$$ This is an equation for dot if we are talking about surfaces, right? It is not an ellipsoid.
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25 views

Find the volume of the region by integration

I have to find the volume of the region bounded by $x + z = 1$;$ y + 2z = 2$;$ x = 0$;$ y = 0 $;$ z = 0$; I tried to sketch the graph separately in the $y-z$ plane and then in $x-z$ plane. But I am ...
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What substitution can be used to evaluate the integral giving the area of the surface of revolution of the curve $x = \sqrt[3]{y}$?

The question is: Find the area of the surface generated by revolving the given curve about the $x$-axis: $$x = \sqrt[3]{y}, \qquad 1 \leq y \leq 8.$$ Now, all is well, simple enough question, ...
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23 views

Prove for criterion that two curve families are orthogonal on a surface in 3D

Let $E, F, G$ be the coefficients of the first fundamental form of a regular surface $R = R(u, v).$ Let $f(u, v) = c$ and $g(u, v) = d$ be two families of regular curves defined in the ...
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58 views

When does $\pi_1(\Sigma)$ inject into $\pi_1(S^3 \setminus \Sigma)$?

Here's a fun fact from knot theory: $\quad$ If $\, \Sigma$ is a minimal-genus Seifert surface for a knot $K$, then $i_*:\pi_1(S^3 \setminus \Sigma) \to \pi_1(S^3 \setminus K)$ is injective, where ...
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26 views

what would the equation of a torus be by making the circunference $(y-2)^2+ z^2 = 0$ and $x=0$ turn along the $z$ axis

What I understand of the question is that I have to, somehow, give the equation of the torus that results of spinning the circumference $$(y-2)^2 + z^2 = 0$$ and $$x=0$$ which as far as I know is just ...
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2answers
17 views

Formula to find lenght of material I need to make a sprint

could somebody write me if you know how to calculate the lenght of a material i need to make a certain spring. I need any true formula you have.
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1answer
17 views

Parameterize the following surface: $x +y +z=1$, $x,y,z>0$.

I need to parameterize the following surface: $x +y +z = 1$, $x,y,z>0$. I tried to put: $\sigma(u,v)=(u+v,u-v,-2u+1)$, but does it solve the case of $x,y,z>0$?
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1answer
22 views

Ruled surface out of lines of curvatures

I'm trying to proof the following statement: Let $c$ be a curve inside a surface element $f:U\rightarrow\mathbb{R}^3$ (i.e $c=f\circ\gamma$ where $\gamma:I\rightarrow U$). Then $c$ is a line of ...
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1answer
57 views

Need help understanding a relation between the fundamental forms

The book I am reading briefly mentions this relation between the fundamental forms but gives no explanation of how they got it. Take the following as the Weingarten Map/Shape Operator where $\nu$ is ...
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24 views

Gradient of second fundamental form

In the book I'm reading ("Differential Geometry Curves-Surfaces-Manifolds by Wolfgang Kuhnel") two definitions of prinicpal curvatures directions are presented: The extramum values of $II(X,X)$ ...
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177 views

What surface is represented by the following equation

$$\sqrt[3]{x^{2}}+\sqrt[3]{y^{2}}+\sqrt[3]{z^{2}}=1$$ Taking cubes of both sides only leads to a more complicated formula. How should one interpret this one. And, also if you could point me to a tool ...
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2answers
68 views

Differential of a rotated f(x, y) surface

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial ...
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52 views

whether any shape can be placed on a tiled surface?

After read "Prove that any shape 1 unit area can be placed on a tiled surface",I think on a surface of equal square tiles where each tile side is 1 unit long,the shape ,less than some constant C>1 ...
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1answer
66 views

$\nabla \times F=0$ implies that $F$ is conservative

Prove that if $F:\mathbb R^3\to \mathbb R^3$ is a vector field so that $\nabla\times F=0$ $\forall x\in \Omega\subset \mathbb R^3$ (where $\Omega$ is an open simply connected set), then $F$ is a ...
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1answer
30 views

“Uniqueness” of the Multi- Dehn-twist

I'm trying to writing down a proof for the following claim about Dehn-Twists: Let $\{a_1,...,a_m\}$ be a collection of distinct nontrivial isotopy classes of simple closed curves in a surface $S$ ...
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1answer
60 views

Surface area generated by revolving $r = \sqrt {\cos 2\theta}$

I've been giving a good time trying to solve this problem, I do not find a clear way to solve appreciate your help. \begin{array}{rcl} r& =& \sqrt{\cos 2\theta } \end{array} This Around to ...
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1answer
80 views

Riemann surfaces with Riemann Roch theorem, linear fiber over an elliptic curve

Let $g:\mathbb{C}\times \mathbb{C^*}\rightarrow \mathbb{C}\times\mathbb{C^*}$ defined by $g(z,w)=(w^n z,\alpha z)$ where $0<|\alpha|<1$. Let $G$ be the cyclic group spanned by $g$ and $A$ the ...
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51 views

Any quartic in $\mathbb P^3$ contains only finitely many lines.

I want to prove thath any quartic $X$ in $\mathbb P^3$ contains finitely many lines, but I don't know any method for computing lines on a surface. What is the idea of the proof?
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27 views

Cornered sphere homemorphic to unit sphere

A buddy of mine asked me this question, to which I found it intuitively obvious, but was unable to come up with a proper proof. Consider the so-called cornered sphere, defined by $x^4+y^4+z^4=1$ in ...
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Can the same surface have minimal genus in both a 3-manifold and a 4-manifold?

By a surface of minimal genus I mean in it's homology class: A surface $S_0$ embedded in a smooth manifold $M$ such that any other surface $S$ with $[S]=[S_0]\in H_2(M)$, we have $g(S)\geq g(S_0)$. ...
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2answers
43 views

How to find the tangent plane to a given point on a surface?

How can you find the tangent plane to a given point on a surface? (Verbal descriptions preferred) I'm thinking you can find the "vector versions" of two directional derivatives (maybe the partial ...
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25 views

Find the “surface vertices” of a collection of points.

I am currently doing some experiments in order to simulate liquids. I have a collection of 3D points that interact with each other to form a body of water. I would like to form a mesh from these ...
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39 views

Shortest smooth paper Möbius Strip

I want to make a familiar Möbius strip of width 1 unit satisfying the physical properties of paper. Assume paper is a ruled surface, and the strip has to be smooth and non-self-intersecting. What ...
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1answer
25 views

Finding all points on a surface where the tangent plane is parallel to the plane 5x+3y-z=0

Consider z=f(x,y)=x^3 + 2xy + y Find all points on the surface where the tangnent plane is parallel to the plane 5x+3y-z=0 So I took the gradient of f (x,y,z) and got (2x^2 i , 2x+1 j , -1 k) And ...
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1answer
36 views

Gauss-Bonnet Theorem, External Angles and Orientation

The Global Gauss-Bonnet Theorem states: Let $R\subset S$ be a regular region and $C_1,\ldots,C_r$ be closed, simple, piecewise regular curves forming the boundary of $R$. Suposse $C_i$ is positively ...
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1answer
16 views

Preservation of the cross product by parametrization

Let $S$ be a regular surface and $X:U \subset \mathbb{R}^2\longrightarrow X(U)\subset S\subset \mathbb{R}^3$ a local parametrization. Does the following hold? If $e_1, e_2$ are two linearly ...
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Intersection of two spherical caps in $(n+1)$-dimensional Euclidian space

I want to compute the intersection area of two spherical caps whose tops are placed on two orthogonal axes in $\mathbb{R}^{n+1}$. In spherical coordinates, the surface element is given by $$ \,dS = ...
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41 views

What is a smooth family of divisors?

Suppose that $S$ is a smooth complex projective surface ($\mathbb C$-scheme, reduced, irreducible...). What do algebraic geometers usually mean with the term a smooth family of divisors in $S$? ...
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54 views

The relationship between Ricci and Gaussian curvatures

Why do we have that for a surface (dimension $2$) that $$\text{Ric}(X, Y) = K \langle X, Y \rangle ,$$ where $K$ is the Gaussian curvature and $X, Y$ are vector fields?
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24 views

Giving parametrization of Hyperboloid

I'd like to give a parametrization of an two sheet hyperboloid, such that this parametrization covers both sheets (Without using $±$ symbols). Does that parametrization exist?
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What Rubik's Twist configuration has the lowest visible surface area?

The Rubik's Twist has been a fun time sink. From the wiki page, [It] is a toy with twenty-four wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that ...
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1answer
39 views

Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
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40 views

Blow up of Hirzebruch surface

I would like to prove that the blow up of $n$-th Hirzebruch surface $F_n=\mathbb P(\mathcal O \oplus \mathcal O(n))$ at a point outside the exceptional section is isomorphic to the blow up of ...
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32 views

About rational curves on elliptic K3 surfaces

I am trying to read this paper http://arxiv.org/pdf/math/9902092.pdf. I have some trouble at the end in Corollary 3.27, and also Proposition 3.24. 0)Morally speaking my main trouble is that i don't ...
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17 views

Orthogonal surfaces

Prove that the three surfaces of the family $xy/z=u$ $\sqrt{x^2+y^2}+\sqrt{y^2+z^2}=v$, $\sqrt{x^2+y^2}-\sqrt{y^2+z^2}=w$ that pass through just one point are orthogonal I´m assuming that first I ...
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22 views

Consider the Plane Curve?

Consider the plane curve $$\gamma(t) = \left( \cosh(t) \cos(t), \cosh(t) \sin(t) \right), \;\; t \in \mathbb R.$$ Is $\gamma$ regular? If $\gamma$ is not regular, can you restrict the parameter ...
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Completing compact surfaces with boundary to closed surfaces in $\mathbb R^3$

My question is whether any compact smooth surface in $\mathbb R^3$ (with smooth boundary) can be completed to a closed smooth surface in $\mathbb R^3$ without boundary? It is easy to complete it to an ...
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1answer
54 views

Curvature of plane curves

What is the neatest way to derive the formula for the curvature $\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $?
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Signed unit normal

I'm trying to study for one of my exams and the past papers have no solutions. I had to define the signed unit normal and the signed curvature. The signed curvature, $\kappa_s$ being such that ...
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Conical surface with negative curvature

I was reading some physics papers and I read about cones possessing negative curvature on the tip (and k = 0 everywhere else). Basically, to build these surfaces instead of removing a sector of the ...
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69 views

How can this be the surface area of an intersection of cone and cylinder?

I have an exercise that requires me to calculate the surface of the intersection between two curves: $Z_1:z^2=x^2+y^2$ and $Z_2:x^2+y^2=2y$. So what I did is the following: Parametrise the cylinder ...
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134 views

A regular surface with non zero mean curvature is orientable [duplicate]

How can I prove that any regular surface with non zero mean curvature is orientable? UPDATE: The surface is embedded in $\mathbb{R}^3$.
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Finding surface area S using area of projection of S??

I was going over my calculus textbook and came across a question about surface area. and question is as follows. Let S be a parallelogram not parallel to any of the coordinate planes. Let ...
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28 views

Re-initialize a 3D spline surface using different control points

I have a 3d spline surface is that is modeled with 65 points where the x,y,z position of each point is known. I want to keep the surface shape, but have different x,y positions for my control ...