Questions tagged [surfaces]
For questions about two-dimensional manifolds.
3,413 questions
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Can a curve in a surface bound an immersed bigon with identical, consistently-oriented boundary labels?
This is a follow up to Can an immersion from the disk to a surface "double up" on its boundary?, to which Lee Mosher gave an illuminating answer that told me I was asking the wrong question.
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Given two points on a unit sphere $p=(p_1,p_2,p_3)$, $q=(q_1,q_2,q_3)$, find the bearing of $q$ from $p$
Given two points on a unit sphere $p=(p_1,p_2,p_3)$, $q=(q_1,q_2,q_3)$, find the bearing of $q$ from $p$.
I have come up with an answer for this problem that I know is wrong. However I don't ...
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Rank of an elliptic surface
I am trying to show that the rational elliptic surface given by:
$$E(\mathbb{Q(}t)): y^2=x^3+(t-1)x - (t^3-3t-1)$$
has rank at least 1.
To do so, by specializing at $t=2$, I get the curve
$$ E_2:y^2=x^...
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Finding a parameterization of the intersection between the plane $n_x x+n_y y+n_z z=0$ and the unit sphere
Find a parameterization of the intersection between the plane $n_x x+n_y y+n_z z=0$ and the unit sphere $x^2+y^2+z^2=1$.
Stuck a little on this
Set the equations equal to each other and rearrange:
$$...
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What properties are guaranteed/required for the support function of a closed, convex 3D surface?
The support function $h(θ, φ)$ of a closed, convex 3D surface gives the signed distance between the origin and a supporting plane which (1) is perpendicular to the vector pointing in the direction ...
- 2,516
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Is there thing such as a perfect spherical surface?
I had a question regarding infinitesimals, multivariable calculus, and integrals.
After recently, looking into deeper meanings of surface, I got a little confused about surface integrations.
My ...
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Smoothness and dimensionality of the zero set of two weighted $p$-norms?
Given $s_1,s_2 \in \mathbb R^n$, $s_1 \neq s_2$, $w_1,w_2 \gt 0$, and $p \gt 1$.
Let
$$
\begin{align}
f_i(x)&= w_i\|x-s_i\|_p,\\
h(x)&=f_1(x)-f_2(x),\\
S&=\{x : h(x)=0\}
\end{align}
$$ ...
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Homotopy vs isotopy of homeomorphisms for surfaces with boundary, punctures, or marked points
Context and Motivation
I am currently studying the mapping class group of surfaces in the context of a course on Teichmüller space and moduli of Riemann surfaces.
In the course, the instructor ...
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Arnold's question on the flux of the vector field $\vec r/r^3$ through a surface
Find the flux of the vector field $\vec r/r^3$ through the surface
$$(x − 1)^2 + y^2 + z^2 = 2.$$
-- Arnold Trivium #12
The answer seems to be $4 \pi$. The divergence is zero everywhere except the ...
- 6,405
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Riemann surfaces diffeomorphic to $\mathbb{C} \setminus \{ 0,1\}$
Consider the smooth manifold $\mathbb{C} \setminus \{ 0,1\}$ or "twice punctured plane" or "thrice punctured sphere". (Note that this is not the space called "pair of pants&...
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Second fundamental form and geodesic curvatures of frame lines
Consider a smooth surface $S$ in $\mathbb{R}^3$ (assuming simple connectivity if necessary) that is parametrized by $\mathbf{r}(u,v)$. With the second fundamental form written as $\text{II}=L\:\mathrm{...
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Is there a geometric way to find the surface area of this part of a channel surface?
A channel surface (a.k.a. canal surface or sphere sweep) is the envelope of a sphere moving along a curve (called the surface's directrix), possibly while varying in radius. I have a channel surface ...
- 2,516
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Prove there is no isotopy interchanging a pair of figure-eight spaces linked with two lines
Consider two figure-eight curves $X$ and $Y$ in $\mathbb R^3$ linked with a pair of straight lines (of infinite extent), as in the image below. How can one prove there is no isotopy interchanging $X$ ...
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Is this an accurate translation from Beltrami's Researches in Analysis Applied to Geometry? [closed]
I have transcribed Beltrami's RICERCHE DI ANALISI APPLICATA ALLA GEOMETRIA. using ChatGPT. I then used ChatGPT to translate it into English.
Consideriamo un sistema di linee a doppia curvatura, ...
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What does Keisler mean by "touch" in his discussion about tangent planes?
I am reading a short section on p. 666 of the third edition of Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler about tangent planes. He defines a tangent plane to a smooth function ...
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