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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Mathematical formula to generate a curved Chinese-style roof

I want to create a Chinese-style curved roof programmatically, something like in the right part of this picture: As seen in the picture, the roof appears to have four curved segments, which intersect at the diagonals. I would appreciate a formula…
Faikus
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Prove that:$\int_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) dx$

I just wanted to ask, if my proof is correct. I haven't seen the equation before, but I think it's quite useful. Let $f$ be an bijective differentiable function. Then the inverse function $f^{-1}$ exists and the following equation…
Rummelluff
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Area of supercircles, or how to integrate $\int_0^1 \sqrt[n]{1-x^n}dx$?

Martin Gardner, somewhere in the book Mathematical Carnival; talks about superellipses and their application in city designs and other areas. Superellipses(thanks for the link anorton) are defined by the points lying on the set of…
Ali
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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 $\gamma$ on a surface $S$ is called a geodesic if…
Mary Star
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A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group classifies closed surfaces. I'd like to get the…
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Intrinsic vs. extrinsic properties of surfaces

I'm reading about the differential geometry of surfaces in $\mathbb{R}^3$. I keep seeing statements about certain surface properties being either "intrinsic" or "extrinsic". Sometimes people say that the intrinsic properties are those that depend…
bubba
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implicit equation for "double torus" (genus 2 orientable surface)

The embedded torus in $\mathbb R^3$ can be described by the set of points in $(x,y,z)\in \mathbb R^3$ satisfying $T(x,y,z)=0$, where $T$ is the polynomial $T(x,y,z)=(x^2+y^2+z^2+R^2-r^2)^2-4R^2(x^2+y^2)$ for $R>r>0$. Is it possible to find a…
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How to show in a clean way that $z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$ is a torus?

How to show in a clean way that the zero-locus of $$z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$$ is a torus?
Damien L
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On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible with the orientation gives us a Riemannian metric…
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How do you prove that the dunce cap is not a surface?

The dunce cap results from a triangle with edge word $aaa^{-1}$. At the edge, a small neighborhood is homeomorphic to three half-disks glued together along their diameters. How do you prove this is not homeomorphic to a single disk?
dfeuer
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Geodesic of a curved surface

I'm trying to read Lambourne's Relativity, Gravitation and Cosmology, but as this seems more of a maths question I've posted it here rather than in the physics forum. The author talks about affinely parameterized geodesics and then, in Exercise 3.9,…
Peter4075
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Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is linearly equivalent to zero? I'm thinking $X\to B$…
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The Gaussian and Mean Curvatures of a Parallel Surface

This is a homework problem from do Carmo. Given a regular parametrized surface $X(u,v)$ we define the parallel surface $Y(u,v)$ by $$Y(u,v)=X(u,v) + aN(u,v)$$ where $N(u,v)$ is the unit normal on $X$ and $a$ is a constant. I have been asked to…
Anri Rembeci
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Surface of the intersection of $n$ balls

Suppose there are $n$ balls (possibly, of different sizes) in $\mathbb R^3$ such that their intersection $\mathfrak C$ is non-empty and has a positive volume (i.e. is not a single point). Apparently, $\mathfrak C$ is a convex body with a piecewise…
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A non orientable closed surface cannot be embedded into $\mathbb{R}^3$

Can someone please remind me how this goes? Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can define the "outward-pointing normal unit vector"…