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Questions tagged [surfaces]

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

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What is the dimension of a hypersurface?

We say a circle is one dimensional and a sphere is two dimensional. More generally, we say the hypersurface $f(x_1,\cdots,x_n)=0$ in $\mathbb{R}^n$ is $n-1$ dimensional. But these hypersurfaces may ...
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Proving that $x^2+2y^2+3z^3 = 1$ is an embedded manifold

I am working on the following exercise: Consider $S = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + 2y^2 + 3z^3 = 1 \text{ and } z>0\}$ Show that $S$ can be parametrised as a graph of a function from an ...
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Regular parametrisations of surfaces and diffeomorphisms

I am stuck at the following exercise: Let $f:D \subset \mathbb{R}^k \rightarrow \mathbb{R}^n$ with $k \le n$ be a regular parametrisation and let $\sigma: \mathbb{R}^k \rightarrow D$ be a ...
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1answer
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Calculate the surface area with integration

Calculate the surface area of the surface obtained when the region enclosed by the given curves is revolved about the $x$-axis $$y=2x^2-8$$ $$y=x^2-1$$ This is a model problem for an exam and I ...
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1answer
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Choose normal to surfaces to enforce their positivity definition

Consider the following: two ovaloids $S$ and $S'$ orientations $N$ for $S$ and $N'$ for $S'$ an isometry $f: S \to S'$ while proving Cohn-Vossen's theorem one contructs the following self-adjoint ...
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Twisted Strip to boundary of surface A\disc\disc is equal A#K\disc

I have struggles to understand the outline of the proof given in "Donaldson-Riemann surfaces". I work with this source: http://wwwf.imperial.ac.uk/~skdona/RSPREF.PDF. On page 21 bottom he mentions the ...
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1answer
24 views

Level curves of exponential

I have the following two-variable function: f(x,y) = exp(-x^2-(y-1)^2) And I need to compute/sketch the level curves for ...
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If a regular surface lie on side of a plane, then the plane is tagent plane

It is exercise 3.35 of Differential Geometry of Curves and Surfaces By Kristopher Tapp I try to prove this exercise by proving: " If a plane passes a ppoint $p$ of a regular surface $S$, and the ...
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2answers
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Equation of tangent plane to a parametrised surface

I've got a problem trying to figure out what I'm doing wrong with these question regarding finding the equation of the tangent plane to a parametrised surface. A surface is parametrised by $$x = u^2-...
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1answer
28 views

Area above a surface

I have to find the area of the surface $$x+y+z=5$$ localizated above the region $x^{2}+y^{2}\leq9$$. I did: $$z=5-x-y\Rightarrow z_{x}=z_{y}=-1$$ So the area is $$\int\int_{R}\sqrt{1+z_{x}^{2}+z_{y}^{...
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convexity of quadratic bounded function

I have a function like this: ${X^T\cdot P\cdot X+Q^T\cdot X-1=0}$ where $\text{X}=\left ( \begin{array}{c} x \\ y \\ z\\ \end{array}\right)$ , $\text{P} = \left( \begin{array}{ccc} \frac14 & \...
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1answer
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Is there an ovaloid that is not topologically equivalent to a sphere?

Topologically speaking, the compact and connected surfaces are classified into three kinds of surfaces: a sphere a connected sum of tori a connected sum of projective planes. Also, we know that: ...
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2answers
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Does the genus $g$ surface for $g\geq 2$ cover the surface of genus 2?

I recently came across a proof that seemed to rely on the fact that $\pi_1(S_g)$ is a subgroup of $\pi_1(S_2)$ for $g\geq 2$, where $S_g$ is the surface of genus $g$. Specifically it was a proof that ...
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Does there exist a regular surface generated by $f(x,y,z)=0$, where $0$ is not a regular value of $f$?

The (geometric) Implicit function theorem states that: If $f:U\subset\mathbb{R}^3\to\mathbb{R}$ is a differentiable function, and $a\in f(U)$ is a regular value of $f$, then $f^{-1}(\{a\})$ is a ...
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1answer
20 views

minimization problem on surface $\Sigma$.

Consider the surface $\Sigma:=\{(x,y,z)\in\mathbb{R}^3:xy-z^2=16\}$. I want to find the point on $\Sigma$ nearest to origin but I have troubles whit this. I define the function $f(x,y)=\sqrt{xy-16}$ ...
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1answer
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Why is the catenoid the minimal surface of revolution?

This is more of a soft question than anything and I'm asking for either a proof or intuitive explanation as to why this is. It seems that if one defines 2 points in the upper-half of the Cartesian ...
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1answer
40 views

theorema Egregium and coeficcients of the second fundamental form

The theorema Egregium says that Gaussian curvature $K$ of a regular surface $S$ is invariant under local isometries. We have a local description of the Gaussian curvature as follows $$K = \dfrac{eg-f^...
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1answer
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Open subset of a regular surface

I have to prove that an open subset of a regular Surface is also a regular Surface I have no clue on How to start that demonstration or what Path to follow
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Show that $K=-\frac{\Delta \log \lambda }{\lambda ^{2}}$

Let $X:\Omega \subset %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$ conformal imersion, $K$ tha ...
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2answers
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Constant Gauss curvature $\Rightarrow$ homogeneous?

Let $S\subset \mathbb{R}^3$ be an embedded surface and $g_S$ the induced metric from $\mathbb{R}^3$ onto $S$. Since isometries preserve Gaussian curvature, $S$ homogeneous $\Rightarrow S$ has constant ...
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$S=\{(x,y,z)\in\mathbb{R}^3:z=f(x,y)\}$ is a surface with a single atlas $(\mathbb{R}^2, Id_{\mathbb{R}^2})$

If $U$ is open in $\mathbb{R}^2$ and $f: U\to \mathbb{R}$ is $C^{\infty}$, then $S=\{(x,y,z)\in\mathbb{R}^3:z=f(x,y)\}$ is a surface with a single atlas$(\mathbb{R}^2, Id_{\mathbb{R}^2})$. I am ...
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Determine if a Bezier surface contains a specific Bezier curve

Let's suppose that we have the following Bezier surface $$ P(t1,t2)= \sum_{i=0}^{3} { \sum_{j=0}^{3} p_{ij} { \varphi _{i}(t1) \varphi _{j}(t2) } } $$ Is there a way to determine if a specific ...
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The derivative of the normal direction passing by a point according to a translation of the point

Suppose given a point $P(x_p,y_p)$ and a curve described by an implicit equation $f(x,y,z)=0$ How to calculate the derivative of the normal direction passing by $P$ according to a translation of the ...
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1answer
122 views

Invariant lifts of a closed curve on a surface of genus > 1.

I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question : Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...
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1answer
146 views

Symmetric monotonic function $f$ on $[0,1]^2$ with $\int_0^1f(x,y)dy=x$

I am searching for an almost-everywhere continuous and monotonic function $f:[0,1]^2\to[0,1]$ with the following properties: $f(x,y)=f(y,x)$ $f(0,y)=0$ and $f(1,y)=1$ (so $f(0,1)$ and $f(1,0)$ are ...
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1answer
63 views

Question on Shortest Path on the Sphere

Consider the following parameterization of the unit sphere: $$X(u,v)=(\sin v \cos u , \sin v \sin u, \cos v)$$ where $u \in (-\pi,\pi),v \in (0,\pi)$. First of all, I am told to find the length of the ...
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1answer
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Nowhere vanishing vector field on Moebius strip

I know there is no continuos non-vanishing normal vector field on Moebius strip, which is pretty obvious. Is it possible to construct a nowhere vanishing tangent vector field on Moebius strip? ...
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Any cylindrical surface of dimension $2$ in $\mathbb{R}^n$ has zero gaussian curvature. What's the geometric intuition behind that?

Here I define cylindrical surface like so: A surface $M^2 \subset \mathbb{R}^{n+1}$ is called cylindrical if there exists a regular parametrization $x(u,v) = (a_1(u), a_2(u), \cdots,a_n(u), b(v))$ ...
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Determine the normal vector to the boundary of a surface

Let $\psi(\theta,r)=(h(r)\cos\theta,h(r)\sin\theta,z(r))$, for $0\leq\theta\leq 2\pi$, $0\leq r\leq L$, be a surface patch for a surface of revolution such that 1.$h'(r)^2+z'(r)^2=1$ for all $r$, 2.$...
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Recommendations for books which covers Surfaces

I'm looking for books which have a good portion on Surfaces. I know Hatcher's Algebraic Topology has a decent part where he does venture into them. But I was looking for something more, surfaces + ...
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Why doesn't conical surface have a stationary (critical) point (at 0,0)?

Function:$$x = {- \sqrt{x^2+y^2}}.$$(a conical surface) To determine whether it has a stationary point or not, 2 condition must be met: function must have partial derivatives at point p0, and ...
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2answers
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Surface with prescribed first fundamental form

Consider a Riemannian manifold $(S,g)$ of dimension 2. What can we say about the possibility of an isometric immersion of this surface into $\mathbb{R}^3$? Of course this is not unique, even up to ...
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Deform surface along some flow

I am designing a software in which the user can cut a surface (such as a sphere or a torus) along some (closed) curve. I would then like the surface to 'unfold' in some way, for example cutting a ...
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1answer
36 views

Product of coefficients of second fundamental form

Let $\mathbb{H} = \{(u,v) \in \mathbb{R}^2| v>0\}$ be the upper half hyperbolic plane with metric $\mu = \frac{du +dv}{v^2}$, and $\sigma : \mathbb{H} \rightarrow S \subset \mathbb{R}^3$ a ...
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Is there an equivalent to trigonometry for solid angles?

Intuitively I would say that it would make no sense, but this question crossed my mind : Is there an equivalent to trigonometry for solid angles?? I haven't found anything yet. Thanks for answering!...
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1answer
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Can two projective surfaces intersect in points only?

Let $S_1,S_2\subset \mathbb P^n$ be two algebraic surfaces (we may assume that $n=3$). Is it possible that $S_1\cap S_2=\{x_1,\dots,x_N\}$ a finite set of points? I can imagine the surfaces two be ...
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1answer
48 views

Moving frames for geometric abstract surfaces

After playing for few time with the method of moving frames for surface in $\mathbb{R}^3$, I decided to try to apply it to study geometric surfaces (topological surfaces with an inner product). I have ...
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Understanding Jacobi matrix in relation to surfaces

For $1 \le k < r$, and $k,n \in \mathbb{N}$ we have a set $ M \subset\mathbb{R_r} $ and a map $\phi : \mathbb{R_k} \supset D \to \mathbb{R_r}$ so that $\phi(D)=M$. Also the Jacobi matrix $J$ of ...
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relationship between surface variance and curvature

I have an explicit surface in 3d, given as map $z=h(x,y)$ ("2.5D") in discrete points on regular grid over $x$-$y$. My goal is to evaluate radius of curvature in each point on the grid. I currently ...
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1answer
47 views

Functions other than $1\over x$ that generate surfaces akin to Gabriel's trumpet

When I took college calculus (more decades ago than I care to admit), we were introduced to a function that would create a surface of revolution having a finite volume but an infinite surface area. In ...
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1answer
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A problem in lines of curvature in T. J. Willmore book

L, M and N are the second fundamental coefficients, and E, F and G are the first fundamental coefficients. Since $\kappa~d\vec r+d\vec N$ is perpendicular to both $\vec r_1$ and $\vec r_2$ I can say ...
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49 views

The level surface of the function $f(x,y,z) = (x^2+y^2)^{-1/2}$ are…

The level surface of the function $f(x,y,z) = (x^2+y^2)^{-1/2}$ are a) Circles centered at the origin b) spheres centered at the origin c) cylinders around the z-axis d) upper halves of ...
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Compact surface with constant mean curvature is a sphere [duplicate]

Someone knows where can I find the proof of the following theorem? A compact surface in $\mathbb{R}^3$ with constant mean curvature is a sphere Thanks in advance
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2answers
28 views

Determine if the intersection of two surfaces is a non-singular curve

Let $S_1$, $S_2$ be two surfaces in $\mathbb{R}^3$ implicitly defined by, respectively, $f_1(x,y,z)=0$ and $f_2(x,y,z)=0$. How do I check if the curve $S_1 \cap S_2$ has singular points without ...
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1answer
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Prove $φ(R^2)$ where $φ(u,v)=(v\cos u,v\sin u,bu)$ is the smooth function.

Prove $φ(R^2)$ is a smooth surface and $φ(u,v)=(v\cos u,v\sin u,bu)$ $R^2\rightarrow R^3$ and b>0 constant. Its rank $Dφ$ is 2, so I'm ok with that part. Only thing to prove is that it has an ...
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1answer
53 views

Translations on flat torus

I'm confused about isometries of the flat 2-torus and could't find anything online that cleared my confusion. My problem is the following: Let $T^2=\mathbb{R}^2/\Gamma$ be a 2-torus for $\Gamma \cong ...
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0answers
39 views

Existence of a relatively flat surface of $\mathbb{S}^3$

Consider $\mathbb{S}^3$ with the standard round metric. Is there an embedded surface $S \subseteq \mathbb{S}^3$ with the following property: $R^{\mathbb{S}^3}(X,Y)=0$ for every two tangent vector $...
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1answer
60 views

Question concerning the surface $\phi(u, v)=(u, v^3, u-v)$

Let $\phi:R^2 \rightarrow R^3$ ,$C^{\infty}$ with $\phi(u,v)=(u,v^3,u-v)$. And $\gamma(t)=(3t,t^6,3t-t^2)$ smooth curve . Prove: $\textbf{a)}$ $M=\phi(R^2)$ is a smooth surface. $\textbf{b)}$ that ...
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1answer
41 views

Prove that $f(M)$ is a smooth surface.

Let $f:R^3→R^3$ diffeomorphism. Prove that for every smooth surface $M \subset R^3$ the set $f(M)$ is also a smooth surface. So to prove that $f(M)$is a smooth surface I need to find a smooth ...
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1answer
36 views

Find the critical values and point of the function.

Exercise. A)$f: \mathbb{R^3} \rightarrow \mathbb{R}$ with $f(x,y,z)=(x+y+z-1)^2$. Find all the critical values and all $c \in \mathbb{R}$ for which $f^{-1}(c)$ is a smooth surface. B)Now let $f(x,y,...