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

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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
22 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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2answers
97 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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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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3answers
58 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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63 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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1answer
75 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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36 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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11 views

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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37 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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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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25 views

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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1answer
40 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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3answers
69 views

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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0answers
28 views

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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1answer
34 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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1answer
37 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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2answers
33 views

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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2answers
49 views

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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1answer
22 views

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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31 views

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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1answer
48 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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1answer
30 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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1answer
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 ...
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15 views

Induced connection on an oriented regular surface $S\subset\mathbb{R}^3$

Let $S\subset\mathbb{R}^3$ be an oriented regular surface - then we have an embedding $\iota:S\rightarrow\mathbb{R}^3$ where $\iota$ is the inclusion of $S$ into $\mathbb{R}^3$. We also have a smooth, ...
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20 views

Slicing a 3d surface using a 2d line equation

So what I'm trying to do is to find the equation of a 2d function on a 3d surface using a 2d line equation. With : $z = f(x, y)$ the equation of the surface and $ax + by + c = 0$ the line ...
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1answer
36 views

Parametrization and area of surface

I have not grasped the way to solve these kinds of problems yet. I need to parametrize the surface and find its area: $S:x^2+y^2+z^2=4$ with $z \ge\frac{\sqrt{x^2+y^2}}{3}$. I have already ...
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4answers
123 views

Parametrizing the surface $z=\log(x^2+y^2)$

Let $S$ be the surface given by $$z = \log(x^2+y^2),$$ with $1\leq x^2+y^2\leq5$. Find the surface area of $S$. I'm thinking the approach should be $$A(s) = \iint_D \ |\textbf{T}_u\times \textbf{T}...
3
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1answer
37 views

How many examples exist of Lie groups that are 2-dimesional surfaces?

It is relatively easy to show that $\mathbb{R}^2$ or $\mathbb{T}^2$ are 2-dimensional surfaces with a structure of Lie groups. I can not find other surface which are also a Lie group, there are more ...
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1answer
41 views

Can we find a regular ($C^k$) parametrization for this surface?

I have here a surface whose curvature properties I want to study, represented in cylindrical coordinates: $$f(r,\theta) = r^2\cos4\theta$$ The problem, however, is that the parametrization is not ...
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1answer
42 views

Triple integral vs double integral to find volume of an object

Is it possible to find the volume of an object bounded by two surfaces in both of these two ways?: -a triple integral of 1 dV (I know this works) -a double integral of the top surface - bottom ...
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1answer
19 views

Showing that, at an elliptic point, a surface lies on one side of the tangent plane.

Let $p\in S$ be an elliptic point of a surface $S$. I want to show that there exists a neighbourhhod $V$ of $p$ in $S$ such that all points in $V$ belong to the same side of the tangent plane $T_p(S)$....
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1answer
53 views

Proof of Euler's Theorem involving curvature.

Theorem: Let $φ$ be the angle, in the tangent plane, measured counterclockwise from the direction of minimum curvature $\kappa_1$ . Then the normal curvature $\kappa_n(φ)$ in direction $φ$ is given by ...
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Proof that the change of parameters between two regular surfaces is a diffeomorphism

I'm stuck on the proof that the change of parameters between two regular surfaces is a diffeomorphism. I'm using Do Carmo's book Differential Geometry of Curves and Surfaces, which can be found online ...
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1answer
29 views

Calculate the Euler-Poincaré characteristic of followin surfaces.

Calculate the Euler-Poincaré characteristic of: An ellipsoid. The surfase $S=\left\{ \left(x,y,z\right)\in\mathbb{R}^{3}:x^{2}+y^{10}+z^{6}=1\right\} $. Note: Not how to do this problem, I not ...
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0answers
10 views

Finding Surface area of a shape given its spherical coordinate equation

Is the surface area of the shape defined by $\rho = 4\cos(\theta)\sin(\theta) $ given by the following? $$\int_0^{2\pi}\int_0^\pi\sqrt{1 + 0 + 16\cos^2(2\theta)}\ \rho^2\sin(\phi)\ \ d\phi\ d\theta$$...
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1answer
35 views

Focal point and principal curvature of a surface

Suppose $S$ is a surface parametrized by $f$ and its Gauss map is denoted by $N$. Define a map $f_t(u,v)=f(u,v)+tN(f(u,v))$. Define a focal point $q$ of $S$ as follows: if there is $t\neq 0$ such that ...
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1answer
32 views

How to present $S_g$ as a $(4g+2)$–gon?

I know we can present $S_g$ (compact surface of genus $g$) as a $4g$–gon with opposite sides identified, but how to present $S_g$ as a $(4g+2)$–gon with opposite sides identified? There is an ...
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2answers
80 views

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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1answer
69 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point $x\...
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1answer
21 views

Prove existence of pair of points on compact surface such that distance is maximized?

Let $M_m$ be a compact $C^1$ surface in $\mathbb{R^n}$. Prove that there exists $x,y,\in M_m$ such that the distance between them is greatest among all pairs on the surface. Then show that the ...