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

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1-dimensional surfaces classification

Hy friends! I need to classify all the 1-dimensional compact surfaces ( in fact, i need those with boundary) and I don't know how to do it. I now the classic books of Guillemin & Pollack or ...
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Maximal offset distance for a surface

Let $\vec r = \vec r(u, v)$ be a regular (analytic) surface. Now we offsetting this surface to distance $d$ in normal direction; new surface is $\vec r' = \vec r + d\vec n$. New surface $\vec r'$ is ...
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What are these quotient spaces homeomorphic to?

I would like to know what the following spaces $X$ and $Y$ look like. More precisely, I want to know if they are homeomorphic to some other known spaces. I define $X$ and $Y$ as a quotient of the ...
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Orientation under local diffeomorphism

Given regular surfaces $S_1$ and $S_2$ such that $S_2$ is orientable and a local diffeomorphism $f: S_1 \rightarrow S_2$, then why is $S_1$ orientable? What I think that can be done is to choose an ...
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Surface element area from constrains

Consider a surface in $\mathrm{R}^n$ defined by $m$ linear constrains: $$\sum_i c_{ki} x_i = 0$$ We assume that the $m\times n$ matrix $c_{ik}$ is full-rank. Then there exists a linear ...
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Understanding how to calculate surface area of parametrized surfaces

I am trying to follow a derivation for surface area of a parameterized surface and my book does not explain the reasoning behind different steps. I understand the derivation for surface area for a ...
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35 views

How to prove the parallel projection of an ellipsoid is an ellipse?

Take the following ellipsoid in implicit form as an example: $$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$ which shows: The parallel projection of the ellipsoid onto $xoy$ coordinate plane can ...
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What is the difference between surface and algebraic curve in general?

The question may seem dumb at first glance. But I couldn't figure out a satisfying answer after some research. A friend of mine told me that in an interview, she was asked to explain the sliding mode ...
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33 views

Volume and surfaces

i need help. Be : $X =(x,y,z)$ and $$T=\{x\in R^3\mid X=\begin{bmatrix}(1+rsin(u))cos(v)\\(1+rsin(u))sin(v)\\rcos(v)\end{bmatrix} ,\\0.5\geq r \geq 0,\\ 2\pi\geq u \geq 0 \\ 2 \pi\geq v \geq0 $$ ...
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1answer
26 views

Is there a simple way to decide if a hyperboloid is one-sheeted or two-sheeted, given the quadric equation?

Let us say that we have a quadric equation, whose solution set lies in $\mathbb{R}^3$, and you know it's a hyperboloid. Is there a way to analytically decide through a criterion if the hyperboloid is ...
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1answer
69 views

Is this spiral known?

Parametrized as $$ \sec \theta \,( p \cos(\theta+ \alpha), \,p \sin(\theta+ \alpha) , c\alpha), $$ the spiral is plotted $ (-\pi/4<\theta< \pi/4;\,\,0< \alpha < 3 \pi) $ for $ p= 1$ ...
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Normal to surface at point

I have this function: $F(x,y,z)=x^2−y^2−z^2+4$ where $z\ge 0,0\le x \le 2,0 \le y \le 2$. How can I find the normal at some point $P=(p_x,p_y,p_z)$? I have tried to calculate the derivatives of ...
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1answer
22 views

surface of the saddle [closed]

i need Help. Determine the surface of the saddle $$S={(x,y,z)∈R^3; x^2+y^2<=2, z=x^2 -y^2}$$ and the flow of $v(x) = x$ , by S plane polar coordinates, dx dy = r dr dφ, are helpful. Thanx
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Absolute value of an RBF distance is less than the absolute value of an actual distance

I have a radial basis function with a linear kernel f(r)=r in 3D. I constructed the surface based on this RBF and noticed that the absolute value of actual distance from any point to the constructed ...
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1answer
58 views

Existence of closed level sets on a surface for some field

Consider an infinite 3D space with only 2 things in it: wind and a solid object. Wind evidently blows around this solid object over its rigid surface. Bascially we are trying to set up a pure field. ...
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What is the value of $\unicode{x222F}_\Sigma \frac{\vec{r} d\vec{S}}{\pi r^3} $

What is the value of $$\unicode{x222F}_\Sigma \frac{\vec{r} d\vec{S}}{\pi r^3} $$ where $\Sigma$ is the "lip" of the region bounded by x+2y+z=6 and $x^2+y^2=(z+2)^2$ x+2y+z=6 is a plane, and $x^2+...
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11 views

Tchebyshef net and Gaussian Curvature $K$

If the coordinate curves form a Tchebyshef net ( Here provides a definition) then $E=G=1$ and $F=\cos(\theta)$. Show that in this case $$K = -\frac{\theta_{uv}}{\sin \theta}$$ When, I calculate ...
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1answer
31 views

Rotaion Surfaces and Complex Numbers

Consider a continuous invertible map $\varphi:\mathbb{R}^+ \longrightarrow \mathbb{R}$, and define the follwing surface $$ s:\mathbb{C} \longrightarrow \mathbb{R} \times \mathbb{C} $$ $$ \qquad xe^{i\...
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26 views

calculate the transition map of a sphere

How can we calculate the transition map between two parametrizations of a sphere, $\sigma(\theta,\varphi)=(\cos\theta \cos\varphi, \cos\theta \sin\varphi , \sin\theta)$, with $U=\left\{(\theta,\varphi)...
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1answer
25 views

Interpolate a rectangular surface with given edges

I need to interpolate a surface by filling a rectangular hole. The height values of the edges are given. I would like to fill the rectangular surface patch by somehow interpolating the edge values. ...
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1answer
24 views

Complement of compact subspace of surface

Let $X$ be a smooth 2-manifold, $K$ be a compact subset of $X$, such that only one component of $X\backslash K$ does not have compact closure, call this component $U$ (there may be other components). ...
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tangent space in a moving coordinate frame

I've got a problem in some geometry of flow. For the sake of completeness I will give the complete derivation of the equation of interest, but I will seperate it into derivation part and question ...
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Surface described by the equation $-3y^2 - 4xy + 2xz + 4yz - 2x - 2z + 1 = 0$

Given the equation : $-3y^2 - 4xy + 2xz + 4yz - 2x - 2z + 1 = 0$. Check if the surface described by that equation has a center of symmetry and then by making the correct coordinate system change, find ...
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Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$

Find the plane which touches the cone $x^2+2y^2-3z^2+2yz-5zx+3xy=0$ along the generator whose direction ratios are $1,1,1.$ Let the plane touches the cone at $(\alpha,\beta,\gamma)$. We know that ...
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59 views

Calculating surface area of intersection between solid cylinder and plane

I wanted to calculate the surface of $$\{(x,y,z) \in \mathbb{R}^3 \mid x^2+y^2<1, x+y+z=1\}$$ but to calculate it, I need a parametrization. My first attempt was to just put: $y=\sqrt{1-x^2}, z =...
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65 views

Surface area of the part of a sphere above a hexagon

I want to calculate the surface area of the part of a half-sphere, which lies above a regular 6-gon. (Radius $r=1$) More formally, Let $G$ be the region on the $XY$-Plane, bounded by the points $\{...
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79 views

Show that the vertex lies on the surface $z^2(\frac{x}{a}+\frac{y}{b})=4(x^2+y^2)$

Two cones with a common vertex pass through the curves $z^2=4ax,y=0$ and $z^2=4by,x=0.$ The plane $z=0$ meets them in two conics which intersect in four concyclic points.Show that the vertex lies on ...
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The section of a cone whose vertex is $P$ and guiding curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,z=0$ by the plane $x=0$ is rectangular hyperbola.

The section of a cone whose vertex is $P$ and guiding curve the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,z=0$ by the plane $x=0$ is rectangular hyperbola.Show that the locus of $P$ is $\frac{x^2}{a^...
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41 views

Top cohomology of a non-orientable smooth surface with boundary.

I would like to know what the singular relative cohomology $H^2(M,\partial M;\mathbb{Z})$ of a smooth connected surface with boundary $M$ is. In the orientable case I did the following: The zero-th ...
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Surface Area of Stainless steel scoop

I want to calculate surface area of Stainless steel Scoop. I am trying different circle and cylindrical formula but not succeeded. please help me out
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How to calculate a surface area of a river?

I am doing a math exploration and I was wondering if someone could help with this problem. What will I need to use in order to calculate SA of a river? What parts of math are used? What info will I ...
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1answer
91 views

If X surface and $ E=1+v^2 $, $ F=0 $, $ G=1 $, $ e=0 $, show that $ a(t)=X(uo,vo+t) $ is a straight curve

Let $X : U \to \mathbb{R}^3$ be a regular surface with $E =1 + v^2$ , $F = 0$ , $G=1$ , $e=0$ Show that the curve $ a(t)=X(uo,vo+t)$ (for constant $ (uo,vo) $ at $ U $ and $ t $ belong at $ (-\...
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How to determine the equation of shortest path on any 3d surface between two given points?

I am working on draping of woven composite and I have to determine the equation of shortest path on 3D surface (i.e. $z=x^2+y^2$) between two given points in order to get the yarn path between two ...
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43 views

Calculate the area of a sphere drilled by two cylinders.

Let $S$ be the sphere given by the equation $x^2+y^2 +z^2 =4$ cut with $z \geq 0$. Now, we drill the semisphere that is left with two vertical cylinders of radius $1$, whose axes are respectively ...
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Find the area between the cylinder $z^2+y^2=r^2$ and two planes

I'm having trouble with this problem: Find the surface area between the top of $z^2+y^2=r^2$ between $z=ax$ and $z=bx$ (consider $a \gt b \gt 0$). I think I must find the area between the ...
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Applications of Banach's fixed point theorem on Differential Geometry

Does anyone know any simple application of Banach's fixed point theorem on Differential Geometry. I am looking for something involving manifolds.f
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35 views

projection of an ellipsoid on XY plane

The equation of an ellipsoid is $$ax^2+by^2+cz^2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0$$ The ellipsoid is arbitrary rotated and the orientation angle are given as θ, β and Ѱ and the center is at (x',y',z')....
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39 views

Computing Gaussian Curvature of surface with polynomial components

I'm studying for a qualifying exam, and I'm working on the following exercise which was on the most recent qualifying exam: Let $\mathbb{R}^3$ have coordinates $(x,y,z)$ and the standard Euclidean ...
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Curvature of curves on surfaces

Are there ways to know the curvature of a curve $\gamma$ that lives on a surface $\mathcal{S}$starting from the gaussian curvature of $\mathcal{S}$? In general, is it possible bound the curvature of ...
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Sketching the surface of $z^2+{(\sqrt{(x^2+y^2)}-R)}^2-r^2=0$

I'm having a problem visualizing the surface $F^{-1}({0})$ where $$F(x,y,z)=z^2+(\sqrt{x^2+y^2}-R)^2-r^2$$ with $0<r<R$ . I have gotten the formula to the point $$z^2+x^2+y^2-2R\sqrt{x^2+y^2}=-...
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Find 2D plane in the center of nonlinear 3D object

I'm building a segmentation algorithm. I'm segmenting pieces of paper in a book that have been slightly crumpled. Imagine taking a piece of paper, crumpling it into a ball, and then trying to ...
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Closed point in the generic fiber of an arithmetic surface

Let $S$ an irreducible Dedekind scheme of dimension $1$, and let $\pi:X\to S$ be a regular, integral fibered surface. We assume that $\pi$ is a flat morphism and that $X_\xi$ is the generic fiber over ...
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Finding the points where a tangent plane is parallel to another plane?

Find all points on the surface ${\bf{r}}(\mu,\nu)=(\mu^2\nu,\mu\nu^2,1)$ where the tangent is parallel to the plane $z=x-y.$ Two planes are parallel if their normal vectors are parallel. That is, ...
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50 views

Presentation of $\pi_1$ of compact orientable surface by induction?

I need to prove by induction $\pi_1(\Sigma_g)= \left\langle a_1,b_1,\dots ,a_g,b_g\mid \prod_i [a_i,b_i] \right\rangle$. For genus 1 this holds since $\pi_1(T^2)\cong \mathbb Z\times \mathbb Z$. For ...
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Surface of the intersection of $n$ balls

Suppose there are $n$ balls (possibly, of different sizes) in $\mathbb R^3$ such that their intersection $\mathfrak C$ is non-empty and has a positive volume (i.e. is not a single point). Apparently, $...
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Method of characteristics - integral surface formation

On page 2 of this PDF from Standord, which describes the Method of Characteristics for first-order PDEs, it is written at the end of the page: "In doing so, we see that $z(x,t)$ is constant along ...
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31 views

Confusion of classification of closed surfaces

I read that we can distinguish closed topological spaces without boundary up to homeomorphism by orientability and euler characteristic - is this correct? But what confuses me is that the Klein ...
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23 views

Connected sum $S_1$ # $S_2$ is commutative and associative

The connected sum of two surfaces $S_1$ and $S_2$ is formed by removing a circular hole from each surface and identifying the boundaries together Show that the connected sum $S_1$ # $S_2$ is ...
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28 views

A specific case of quadratic forms

I have a quadric as follows: $$ax^2+by^2+bz^2+yz=0.$$ I am curious to know which shapes in $\mathbb{R}^3$ this equation describes for different value of $a$ and $b$?
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Surface integral of a scalar over a unit cube.

Evaluate the following integral $$\iint_S (x+y+z) \, dS$$ where $S$ is the surface of the cube $[0,1] \times [0,1] \times [0,1]$ Honestly, I don't know what to do. All I know is that you have to ...