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

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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 arctan in ...
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Identification space of square. Net, triangulation and surface classification

Space Z is made as an identification space of unit square $Q=${$(x,y) | 0\leq x, y \leq 1$} by making the following identifications: $ (0,y)$~$(1,y) $ for all $0\leq y\leq 1 $, $ ...
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Surface Normal To Euler Angles

I am working on an application to extract positions from a point cloud. My point cloud has three axis X, Y, Z I am using PCA to generate a surface normal from a section of a surface so I end up with ...
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Examples of surfaces

I have to find an example of a surface of revolution excluding a sphere and a cone. Is $\sigma(x,y)=(\cos x, 5, x^2+y^2)$ such an example? $$$$ I also have to find an example of a surface the ...
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Surface Area for a Curve Rotated Around the Y-Axis

I tried finding the surface area of a function rotated about the y-axis but I don't trust my answer. If I am looking for the surface area of a function y=f(x) rotated about the y-axis. $$S= 2\pi ...
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Laplace-Beltrami of the Gauss map

I'm looking for the proof of very nice identity about the Laplace-Beltrami operator of the Gauss map $N$ of a regular surface in $\mathbb{R}^3$ given by a patch $X$. I want to show that $$\Delta N = ...
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1answer
46 views

Condition to be conformal

I am looking at the following exercise: Let $\Phi : U \rightarrow V$ be a diffeomorphism between open subsets of $\mathbb{R}^2$. Write $$\Phi (u, v)=(f(u, v), g(u, v))$$ where $f$ and $g$ are ...
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51 views

The parameter curves are asymptotic curves

I am looking at the following exercise: Let $p$ be a hyperbolic point of a surface $S$. Show that there is a patch of $S$ containing $p$ whose parameter curves are asymptotic curves. Show that ...
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70 views

Is this a misprint in Do Carmo's 'Curves and Surfaces'?

I'm reading the following section from the book 'Curves and Surfaces' by Do Carmo, but I'm stuck and after having gone over this like 10 times I'm starting to think it must be a misprint. The problem ...
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67 views

Which is the intersection?

I am looking at the last question of the following exercise: $$$$ Which exactly is the intersection of any surface from one family of the triply orthogonal system with any surface from another ...
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69 views

Equation of a cone

Find the equation of the cone whose vertex is at the origin and whose directing curve is given by the equations: $$\begin{cases} x^2-2z+1=0 \\ y-z+1=0\end{cases} $$ We know that an eliptic cone is ...
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3answers
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Orientable surface

Suppose that two smooth surfaces $S$ and $\tilde{S}$ are diffeomorphic and that $S$ is orientable. I want to prove that $\tilde{S}$ is orientable. $$$$ Since $S$ and $\tilde{S}$ are ...
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How to minimize the surface area taken by a cylinder?

In my math class, we are working on Geometric Optimization problems. We have to create an equation, and then solve for one variable, in terms of another variable. Then, using an expression, we find ...
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1answer
68 views

Topological surface covered by hexagons and heptagons

I've found an interesting exercice that I don't know how to approach. It goes like this. We have a topological space which is Hausdorff, compact, connected and locally homeomorphic to ...
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1answer
78 views

The straight lines are contained in $S$

I am looking at the following exercise: The hyperboloid of one sheet is $$S=\{(x,y,z)\in \mathbb{R}^3 \mid x^2+y^2-z^2=1\}$$ Show that, for every $\theta$, the straight line $$(x − z) \cos \theta = ...
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28 views

How could we show that these are perpendicular?

I am looking at the following exercise: Suppose that the first fundamental form of a surface patch $\sigma (u, v)$ is of the form $E(du^2 + dv^2)$. Prove that $\sigma_{uu} + \sigma_{vv}$ is ...
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26 views

Is this (self intersecting) surface considered one sided?

I wanted to apply stokes theorem on a curve (in black) that possibly looks like the seam of a tennis ball. I make a surface by drawing a line from the origin to each point of this curve. I get a ...
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1answer
97 views

The surface is an open subset of a sphere

I am looking at the following exercise: $$$$ Could you give me some hints how we could show that? Do we use the matrix of the Weingarten map with respect to the basis $\{\sigma_u,\sigma_v\}$ ...
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How can we find geodesics on a one sheet hyperboloid?

I am looking at the following exercise: Describe four different geodesics on the hyperboloid of one sheet $$x^2+y^2-z^2=1$$ passing through the point $(1, 0, 0)$. $$$$ We have that a curve ...
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52 views

Equation of a 3D curve shaped like a logarithmic spiral

I'm not exactly certain of the mathematical description of this surface (if I were I wouldn't have a question), but I basically want to make a "3D spiral" which is basically a sine wave "wrapped" ...
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Reference to an atlas of curves and surfaces?

I remember at more than one university math department there being a set of glass cabinets with a number of physical models of surfaces. They were all algebraic varieties on the reals (of limited ...
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What's the name of the surface and Is it a $C^2$ smooth surface? [duplicate]

what's the name of the surface? Is it a $C^2$ smooth surface? Its implicit equation is: $(x−2)^2(x+2)^2+(y−2)^2(y+2)^2+(z−2)^2(z+2)^2+3(x^2y^2+x^2z^2+y^2z^2)+6xyz−10(x^2+y^2+z^2)+22=0$
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maximum number of faces with n lines

I was wondering a formula F(n) to guess the maximum number of faces made with n lines, for example: with 1 line, we cant create a face; F(n) = 0; with 2 lines, we also cant create a face F(n) = 0; ...
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Unit normal of the surface $S$

We have that $$\sigma (u,v)=\gamma (u)+v\delta (u)$$ and $$K=\frac{-(\dot\delta \cdot \textbf{N})^2}{EG-F^2}$$ I want to show that if $\gamma$ is a curve on a surface $S$ and $\delta$ is the unit ...
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34 views

When does there exist a isometric transform between the surfaces $S$ and $\widetilde{S}$?

Suppose there are two $E^3$ surfaces, $$S:\mathbf{r}(u,v)=(au,bv,\frac{au^2+bv^2}{2})$$ ...
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How can we calculate the unit normal $\textbf{N}$ of the sphere?

I want to show that the normal curvature of any curve on a sphere of radius $r$ is $\pm \frac{1}{r}$. $$$$ The normal curvature is $\kappa_n=\gamma '' \cdot \textbf{N}$, where $\gamma$ is a ...
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65 views

Compact surface

To check if the surface $x^2-y^2+z^4=1$ is compact, we have to check if the surface is closed and bounded. Could you give me some hints how exactly we check that? How can we check if it closed and ...
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91 views

Revolution of fractal

How to find the volume and surface area of a shape which made from revolution of Koch Snowflake? (I think the surface area will be an infinity, because length of the Koch snowflake is infinity.) And ...
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72 views

Open subset of a plane [duplicate]

Suppose that the second fundamental form of a surface patch $\sigma$ is zero everywhere. How can we prove that $\sigma$ is an open subset of a plane? The second fundamental form of a surface patch ...
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128 views

Two twisted cubic curves in $\mathbb P^3$ intersect iff they lie in a common cubic surface

Let $C_1$ and $C_2$ be twisted cubic curves in $\mathbb P^3$. I want to prove that they intersect if and only if they lie in common cubic surface, perhaps singular. The second condition can be ...
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109 views

Show that it is a smooth surface

Show that the ellipsoid $$\frac{x^2}{p^2}+\frac{y^2}{q^2}+\frac{z^2}{r^2}=1$$ where $p$, $q$ and $r$ are non-zero constants, is a smooth surface. To do this do we have to take a parametrization of ...
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Why is this the equation of the tangent plane?

I want to find the equation of the tangent plane of the surface patch $\sigma (r, \theta)=(r\cosh \theta , r\sinh \theta , r^2)$ at the point $(1,0,1)$. I have done the following: The point ...
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39 views

Visualizing order 3 mapping class of genus 2 surface

Let $\Sigma_2$ be a closed genus $2$ surface. There exists an orientation-preserving diffeomorphism $f:\Sigma_2 \rightarrow \Sigma_2$ of order $3$. The diffeomorphism has $4$ fixed points (each, of ...
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Hypersurfaces containing given lines

Let $L_1, ..., L_n$ be non-intersecting (general, if necessary) lines in $\mathbb P^3$. I need to find the dimension of the space of polynomials of degree $d$, vanishing on these lines. ...
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Why is $\pi_1(F_g)^{ab} = \Bbb Z \langle a_1, b_1, \ldots , a_g, b_g \rangle$?

I am told that for a surface with genus $g$, call it $F_g$, the abelianization of $\pi_1(F_g) = \langle a_1, b_1, \ldots , a_g, b_g \mid [a_1, b_1] \cdots [a_g,b_g] = e \rangle$ is $\pi_1(F_g)^{ab} = ...
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33 views

Geodesics on the Surface of Revolution from do Carmo's book

This is a question I encountered at DoCarmo's Differential Geometry of Curves and Surfaces p258. I do not know this sentence just below the second equation: "(Of course the geodesic may be tangent ...
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Determine the dimension of the set of surfaces of $\mathbb{P}^{3}$ that contain certain conic.

Let $C\subseteq\mathbb{P}^{3}$ be the conic of equations $$ C=V(X_{3}, X_{0}X_{2}-X_{1}^{2})=\{(t_{0}^{2}:t_{0}t_{1}:t_{1}^{2}:0)\in\mathbb{P}^{3}:(t_{0}:t_{1})\in\mathbb{P}^{1}\}. $$ I have to ...
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125 views

Curl of unit normal vector on a surface is zero?

I have a scalar field $\phi$. From this field, I define an iso-surface $\phi=\phi_{iso}$. The unit normal vector on this surface is ...
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Surface of a torus in terms of Legendre polynomials

The equation of a spheroid is $$\frac{x^2 + y^2}{a^2} + \frac{z^2}{b^2}$$ Its surface can be expressed as $$ r = a \left( 1 - \frac{2}{3} \epsilon P_2(\cos \theta) \right) $$ where $r$ is the ...
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Applications of the generalized Gauss-Bonnet Theorem for surfaces

Quoting Do Carmo's 'Differential Geometry of Curves and Surfaces': "We have only to think of all possible shapes of a surface homeomorphic to a sphere to find it very surprising that in each case the ...
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surface has two geodesic with fixed angle must be a developing surface

It seems easy to show by Gauss-Bonnet theorem that a surface which has two families of geodesics with fixed angle $\theta$ must be locally flat, i.e. its Gauss curvature $K=0$. But to show it is in ...
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Gauss-Bonnet-Chern Theorem coincides with Gauss-Bonnet Theorem in the two dimensional case.

The generalized Gauss-Bonnet theorem says that: Let $M$ be a closed oriented Riemannian manifold with an even dimension $n$, then $$ \int_{M}\Omega=\chi(M). $$ In that formula, $\chi$ is the euler ...
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Surface areas of different submanifolds of $\Bbb R^3$

Can you circumscribe a continuous, smooth manifold in $\Bbb R^3$ with another manifold that completely encapsulates it but has a surface area which smaller than that of the one contained? Is there a ...
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Must $\vec{n}$ be a Unit Normal Vector (Stokes' Theorem)?

If $S$ is an oriented, smooth surface that is bounded by a simple, closed, smooth boundary curve $C$ with positive orientation, then for some vector field $\vec{F}$: $$\oint_C \vec{F} \cdot d\vec{r} ...
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Parameterization of Ellipsoid

I have a question asking me to evaluate $\iint_\Sigma \mathbf{F} \cdot \mathbf{n}~dS$, where $\Sigma$ is the lower half of the ellipsoid $z = -2 \sqrt{1 - x^2 - y^2}$ with $\mathbf{n}$ directed ...
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30 views

Parametrization of a cylinder that is parallel to x axis

The answer is no it does not matter. The surface is $y^2+z^2=4$, I parametrized it so: $\mathbf r=x \mathbf i +2\cos\theta \mathbf j + 2\sin\theta \mathbf k$ But Pauls Outline works through the ...
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31 views

Evaluation of an oriented surface integral

I am having a bit of trouble understanding example 1 in Paul's Calculus Notes page on surface integral: http://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx I understand how to do ...
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Decomposing a surface $S$ with a simple closed curve $\Gamma$

In class we learned about how the Euler characteristic changes when we take a connected sum of surfaces $M_1$ and $M_2$: $$\chi(M_1 \# M_2) = \chi(M_1) + \chi(M_2) - 2,$$ and it made me wonder how ...
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40 views

What exactly is a surface integral?

I'm learning surface integrals right now and I don't think I fully understand what they are. What exactly do surface integrals represent? Is it volume? The basis for surface integrals seems just like ...
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50 views

How big are regular (hyperbolic) polygons?

Given a hyperbolic surface of constant curvature $K=-1/a^2$ embedded in $\mathbb{R}^3$, is there a known formula for the length of the edges of a regular polygon? I know that the Gauss–Bonnet ...