Questions tagged [surfaces]

For questions about two-dimensional manifolds.

Formally, a surface is a two-dimensional topological manifold. Some examples of surfaces are the plane, the cylinder, the sphere, and the graph of a real-valued function of two variables.

More generally, the term "hypersurface" can be used to denote an $(n-1)$-dimensional submanifold of an $n$-dimensional manifold.

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Estimate the number of connected componets of a compact surface $S$ in terms of the number of critical points of a function defined on $S$

Consider the following problem (Exercise (5) at the end of Chapter 2 of Curves and Surfaces, 2nd Edition, by Montiel and Ros): Take a compact surface $S$ and a differentiable function $f: S \longrightarrow \mathbb{R}$ defined on it. Estimate the…
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If $S$ is a compact surface and $R$ is a straight line, then there is some $p$ in $S$ such that the normal line intersects $R$ perpendicularly

Consider the following problem (Exercise 2.71 in Curves and Surfaces, 2nd Edition, by Montiel and Ros): Let $S$ be a surface and let $R$ be a straight line of $\mathbb{R}^3$. Prove that if $S$ is compact, then there is a point of $S$ whose normal…
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Prove that normal lines to a cylinder cut the axis perpendicularly

Consider the following problem (Exercise 2.55 in Montiel and Ros's Curves and Surfaces, 2nd Edition): Let $S = \{p \in \mathbb{R}^3 \ | \ |p|^2 - \langle p, a \rangle^2 = r^2\}$, with $|a|=1$ and $r>0$, be a right cylinder of radius $r$ whose axis…
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Is a pentagon a surface?

My question arise since a pentagon is homeomorphic to a closed disc. This last one is a surface with boundary. However, a pentagon has vertices, so it seems isn't a 2-manifold. If you consider a pentagon without edges, it is a 2-manifold, however…
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How to find a surface if we know the normal vectors?

I have the positions of 3D particles at each z-section each color represents a particle. The lateral projection of the data looks approximately like this imageThe top end of these particles form a curved surface I want to generate a surface locally…
Elizabeth
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Uniform Distribution of Points on a Surface in $\mathbb{R}^n$

Let $\phi : U \subset \mathbb{R}^k \to \mathbb{R}^n$ be an embedding of a $k$-dimensional surface in $\mathbb{R}^n$. Is there a general prescription for selecting points $p$ in $U$ such that the points $\phi(p)$ will be uniformly distributed in…
Charles Hudgins
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Prove that the cone given by $σ(u,v) = (\cos(u)v, \sin(u)v, v)$ with $0 < u < 2\pi$ and $0 < v < \infty$ is isometric to (part of) the plane

Prove that the cone given by $σ(u,v) = (\cos(u)v, \sin(u)v, v)$ with $0 < u < 2\pi$ and $0 < v < \infty$ is isometric to (part of) the plane. Ok so far I have the first fundamental form $\sigma$ which is $ds^2$ $= v^2 du^2 + 2dv^2$ However, to my…
mathsnerd22
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How can we parametrize the following surface?

How to parametrize the following surface in $\mathbb{R}^3$: the intersection of $S=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2\leqslant 1\}$ and $D=\{(x,y,z)\in\mathbb{R}^3:x+y=1\}$. Any hints are welcome.Thanks!
Kato yu
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How do the fundamental forms change under an image of a diffeomorphism?

Let $M \subset \mathbb{E}^3$ be a surface, $r:U \to M$ be its local parametrization, $f: M \to N \subset \mathbb{E}^3$ a diffeomorphism of manifolds. This induces a parametrization of $s: U \to N$. The parametrizations $r, s$ induce bases of the…
felyaah
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What is the local canonical form of a regular surface?

We know that for any space curve $\alpha:I\rightarrow {\rm I\!R}^3$ parametrized by arc length, there exists a "local canonical form" around $s_0\in I$: $$ \alpha(s) = \begin{bmatrix}s-\frac{k^2 s^3}{6} + o(s^3)\\ \frac{k's^3}{6}+\frac{s^2…
T.L
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Gradient as normal to a surface

Why the normal to a surface is given by gradient? How to see this intuitively?
danny
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Surfaces in Three Dimension

This is from Pressley’s book. This theorem gives conditions under which a level surface is a smooth surface. Here $S$ is the given surface and $P$ a point on $S$. My question is what is the meaning of (i)? For a surface patch we always take an open…
danny
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Formula for the solid angle subtended by a surface using the divergence theorem

Let $K$ be a compact subset $\mathbb R^3$ with a $C^1$-boundary, i.e. for all $x\in\partial K$ there is an open neighborhood $U$ of $x$ and a $\psi\in C^1(U)$ with $$K\cap U=\left\{u\in U:\psi(u)\le0\right\}\tag1$$ and $$\psi'(u)\ne0\;\;\;\text{for…
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How is the normal field for a 2-dimensional submanifold of $\mathbb R^3$ defined?

Let $M$ be a 2-dimensional embedded $C^1$-submanifold of $\mathbb R^3$, i.e. a subset of $\mathbb R^3$ such that for all $x\in M$ there is a 2-dimensional chart$^1$ $(U,\phi)$ of $M$ with $x\in\phi(U)$. In particular, for all $u\in U$ and…
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Slicing Level Surfaces

How do I properly do questions like these? I simply each equation by inputting the value into the variable but after that I am confused as to what I am supposed to do after?
OGK
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