Mathematics Stack Exchange
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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.

3209 questions
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Natural realizations of closed orientable surfaces

A beautiful fact is that The space of configurations of a 5-vertex polygon with unit length sides, two of whose vertices are fixed, is a closed orientable surface of genus $3$. Similarly, but much more simply, the torus is the space of…
Mariano Suárez-Álvarez
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How do I find the area of a triangle, in 3D, that lies between two planes, z = A and z = B

Very simple problem to conceptualize, but I don't have a good mathematical solution. I have a triangle with P0 = (x0, y0, z0), P1 = (x1, y1, z1), and P2 = (x2, y2, z2). The triangle represents part of a surface on a map. Furthermore, if the…
Brian West
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Is there a common name for the surface z = xy?

I would call it a saddle, but it's not the standard saddle. Is there a standard name for it, the way we have 'hyperboloid of one sheet' for example?
Henry
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What is the object on the front of Larson and Edwards' calculus and pre-calculus textbooks called?

There is this incredible glass figure on the front of my Calculus textbook, I searched online for what this figure is called and the formula for creating it, but I can't find it. I think it is a variation of the Klein Bottle. Here is the image:
user249952
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Boundary of Mobius strip is $S^1$

I feel like this should be simple, and it is intuitively obvious by looking at the polygon with side identifictations version of the Mobius band, but how do we explicitly show, i.e find the homeomorphism, that the boundary of the Mobius band is the…
Wooster
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When is the Gauss Map on a surface bijective?

If $\mathcal{S}$ is a smooth, compact hyper-surface in $\mathbb{R}^n$ with positive definite second fundamental form, can we say that its Gauss map is bijective? If so, why? Thanks!
CeCe
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Why is the dual tree to a measured geodesic lamination in a compact hyperbolic surface not complete?

Let $M$ be a closed connected surface and $\mathcal{F}$ a minimal (every leaf is dense) measured foliation (as, for example, in Thurston's work on surfaces) on $M$. Let $\tilde{M}$ be the universal cover of $M$ and $\tilde{\mathcal{F}}$ the pullback…
ah--
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I can find slices of a surface for any angle through the vertical axis, but I am not certain what the equation of the surface is.

I can find slices of a surface for any angle through the vertical axis, but I am not certain what the equation of the surface is. Equation along x-axis: $z=1/(1+x^2)$ Equation along y-axis:$z=1/(1-y^2)$ Equation at 45 degrees: $z=1/(t^4+1)$, where…
User3910
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Regular Surface ("8"-cylinder )

Let $C$ be a figure "8" in th $xy$ plane and let $S$ be the cylindrical surface over $C$ that is, $$S=\{(x,y,z)\in \mathbb{R}^3; (x,y)\in C\}.$$ Is the set $S$ a regular surface ? So my answer is no because based on the proposition 2 in Do Carmo…
Bernstein
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What is the equation of the plane?

Suppose a plane has a unit normal vector of $\left(\frac {1}{\sqrt6} , \frac {1}{\sqrt6} , \frac {2}{\sqrt6}\right)$ and is at a signed distance of $\left(\frac {13}{\sqrt6}\right)$ from the origin. What is the equation of the plane? So given $d$ =…
confusedbeginner
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Find the intersection of two surfaces

I have been looking into this question : we have two surfaces : $$\big\{(x,y,z)\in \mathbb{R}^3 \mid\;\; S_1\colon\;\; x+z=1 ,\;\; S_2\colon\;\; x^2+y^2=1 \big\}$$ we need to draw or describe the "shape" that we get . I tried to solve it by drawing…
Waseem Francis
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How to plot 3D figures correctly?

When we always draw e.g. cylinder on the whiteboard, we get kind of this: But naturally (assumed axes are perpendicular) when we have z-axis to the top and y-axis to the right, x-axis must point to us (from display) like this: And when we look…
nakajuice
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Can any 3d surface be mapped to a 2d plane?

I could see a 3D surface being parameterized by 2 parameters such that the surface is broken up into many infinitesimally thin curves. Is this true even for surfaces that might "fold" over itself or curve back to connect to itself? e.g. a cylinder…
Alex
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A lemma on the growth of the number of certain edge paths for a given train track

How to prove the following lemma from the book "Closed curves on surfaces" written by Francis Bonahon? Lemma: For any fattened train track $\Phi$, the number of edge paths of $\Phi$ of length $r$ that are followed by embedded arcs grows…
Yicky
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Generation of a formula to map onto response surface analysis values/predict its maximum

I'm new to StackExchange, so sorry if I don't format this question amazingly. I'm currently performing research on how the use of different proteins in a cell scaffold can change how well the scaffold promotes cell proliferation. Here's my setup: X:…
BuyLegacy
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