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

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Cylinder with two puncture and one cone point

I have questions about a Cylinder with two puncture and one cone point of order $3$, is it a singular pair of pants or there is other possibility?? This surface is not compact as it has two puncture, ...
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1answer
26 views

Area of ​the surface of revolution of the ellipsoid

I need to find the surface area of an ellipsoid using the equation of an ellipse. I believe my calculations are correct but the formulas I meet on the Internet are complex and have $\arcsin$ or $\...
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33 views

Determine all points at which the surfaces share the same tangent line.

Determine all points at which the surfaces $x^2+y^2+z^2=3$ and $x^3+y^3+z^3 =3$ share the same tangent line. I know how to get the same tangent line for the curves, but I'm not sure how to go about ...
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2answers
33 views

How do you find the metric tensor for a given manifold?

Is there some general way to derive the metric tensor for a given manifold M? For example, how was the metric for the surface of a sphere $$ds^2=d\theta^2+\sin^2\theta \, d\phi^2$$ first derived?
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1answer
17 views

Parameterize $\{(x,y,z) \in \mathbb{R}^3 \colon (\sqrt{x^2 +y^2} -3 )^2+z^2 = 1\}$

I need to parameterize the surface $$S=\{(x,y,z) \in \mathbb{R}^3 \colon (\sqrt{x^2 +y^2} -3 )^2+z^2 = 1\}.$$ My hint is that $S$ is a torus. I barely know where to begin. I have some idea on perhaps ...
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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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21 views

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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1answer
25 views

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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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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19 views

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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62 views

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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1answer
485 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves $\...
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2answers
74 views

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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1answer
89 views

Questions about surface integrals and an example problem

It is double integral $(x + y) dS$ where $S$ is the part of the cylinder $y^2 + z^2 = 4$ . With $x$ being between $0$ and $5$ First question, if we want to get the integral of the surface of a ...
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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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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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2answers
28 views

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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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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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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466 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...
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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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2answers
38 views

Fast Rational Bézier Surface Evaluation Problem

I am currently writing a NURBS ray tracer. What I do is convert the NURBS into rational Bézier patches and then perform the intersection test using Newton's method. To do this fast (the ray tracer ...
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44 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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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
25 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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515 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
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3answers
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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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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99 views

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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889 views

Identify and sketch the quadric surface?

I'm stuck trying to figure out which type of quadric surface this equation is: $$\dfrac{x^2}{16} - \dfrac{y^2}{9} - \dfrac{z^2}{1} = 1$$ I have narrowed it down to a hyperboloid, but cannot ...
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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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1answer
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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Triangulation of torus - understanding why

Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider simplicial complexes. I do not understand why the triangles ...
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0answers
28 views

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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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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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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1answer
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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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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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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41 views

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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Find surface area that lies above a triangle

Determine the area of the part of the surface $z=2 + 7x + 3y^2$ that lies above the triangle with vertices $(0,0)$, $(0,8)$, and $(14,8)$. I do not know what formula to use to attempt this problem!
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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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2answers
315 views

What equation will create a 3D rose curve?

The parametric equation $x=a\cos(bt)\cos(t)$, $y=a\cos(bt)\sin(t)$ where $a$ & $b$ are constants and $t$ is parameter gives a rose curve which looks like, On a similar basis, is there a equation ...
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1answer
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 ...