Questions tagged [surfaces]

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n-sided surfaces

Does anyone of you know a good representation of n-sided surface with multiple loops.(I am building an app.) I knew the edges of the face. The edges are 3D curves. It also would be very good if the ...
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Degree two map from genus two surface to genus one surface

I am constructing a degree $2$ map from the genus-two surface $S_2$ to the genus-one surface $S_1$. Searching on this website, I noticed the following approach: Let $\Sigma$ be a compact connected ...
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Showing that surface area is equivalent to $\int_{S}\|\partial_u\phi\times\partial_v\phi\|dudv$, and is there MVT for bijections: $\Bbb R\to\Bbb R^2$?

$\newcommand{\d}{\,\mathrm{d}}$It can be shown that arclength, considered as a sum of increasingly fine partitions of the graph, approaches the integral formulation. However, I have only ever seen the ...
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How to tackle level curve problems

My professor always asks level curve problems on the exam, and I can't seem to answer them right, every time I think I get them they ask something weird, let's note the last attempt where they asked ...
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Example of a fundamental polygon

I was studying some Topology and learned to the concept of fundamental polygons and their surface equivalences. Playing with some configurations I couldn't make sense of the following one. Gluing the ...
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End space of non-compact 2-manifolds in terms of proper rays

I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. Below I ...
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Cohomology of surfaces without homology

I'd like to find a book or other source with a detailed calculation of the cohomology of the connected sum of n torus and of the connected sum of n real proyective planes. It can assume that you ...
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For a fixed surface area, what is optimal shape of a boat so that it can carry the most weight?

This problem is motivated by the Penny Boat Challenge: you are given an aluminum foil and you have to create a boat out of it that can hold the most amount of pennies. I know that Archimedes' ...
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Contradiction with the dimension of shape operator matrix

Context My question is about the matrix dimension of the shape operator. In order to avoid misunderstanding let $S \subset \mathbb{R}^3$ be a regular surface and $$\psi(u,v)=(x(u,v),y(u,v),z(u,v))$$ ...
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Locally Riemannian Homogeneous Surface

A Riemannian manifold $M'$ is Riemannian homogeneous if $Iso(M')$ acts transitively. A Riemannian manifold $M$ is locally Riemannian homogeneous if there exists a Riemannian homogeneous manifold ...
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Are principal congruence subgroups ever surface groups?

Define the principal congruence subgroup of level $n$, $\Gamma(n)$, to be the subgroup of $PSL_2(\mathbb{Z})$ which is congruent to the identity mod $n$. $\Gamma(n)$ is discrete infinite, ...
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Surface area between two known 3D polynomials

I have two curves in 3D space with known equations of the form z = ax + by. Curves I find the coefficients a and b with some simple Python scipy curve-fit code and the equations are well fitted. My ...
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Explication of the differential of a smooth map

I am new on the topic of differential of a smooth map and was looking for some help. I observe a surface $S$ and look at the following smooth map: $$f: S \to S$$ Now I am looking at $df$, hence his ...
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Oriented surface, oriented curve, and vector field

I have an oriented surface S and an oriented curve C. The surface and the curve intersect in points A and B. In A, the orientation of the curve is the same of the surface, in B they are in opposite ...
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Non-trivial Examples of Surfaces of Voss

Let $I$ and $J \subset \mathbb{R}$ be two intervals of the real line. A smooth parametrized immersed surface $\sigma: I\times J \rightarrow \mathbb{R}^3$ is called a surface of Voss if its coordinate ...
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Gaussian curvature as a smooth function

Why is it that for a regular surface S in $\bf{R}^3$ the Gaussian curvature is a smooth function over S? Does this also hold true for $\bf{R}^n$? Intuitively it makes sense, but I am having trouble ...
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Normal unit vector in cylindrical coordinates

Suppose I have a surface in cylindrical coordinates given by $z=f(r,\theta)$. How can I proceed to find the normal unit vector of this surface? My initial guess was to evaluate it's gradient, which ...
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Theorem VI in GENERAL INVESTIGATIONS OF CURVED SURFACES by Gauss

I am reading Gauss's paper GENERAL INVESTIGATIONS OF CURVED SURFACES, but encountered a problem in Theorem VI (p. 4). How does the following observation come from? In Gauss's notation, $LL'$ denotes ...
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Is a closed/open nondegenerate triangle a submanifold (with boundary)?

Let $M\subseteq\mathbb R^3$ be a (nondegenerate) closed/open triangle spanned by $p_0,p_1,p_2\in\mathbb R^3$. Can we show that $M$ is a $2$-dimensional embedded $C^1$ of $\mathbb R^3$ (possibly with ...
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Is there always a complete graph of maximum chromatic number?

The Four Colour Theorem states that the maximum chromatic number of a planar graph is four. The upper bound is achieved by $K_4$. Two ways to generalise this theorem which can be combined are to ...
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How to project a parametric curve to a free-form parametric surface

I need to project a parametric curve $\vec{r(t)} = (r_x(t), r_y(t), c)$, where $c$ is constant, into a free-form parametric surface $\vec{S(u,v)} = (S_x(u,v), S_y(u,v), S_z(u,v))$. However I did not ...
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A combinatorical approach to classical Riemann-Roch

I am reading "Riemann-Roch and Abel-Jacobi Theory on a Finite Graph" by Baker and Norine (2007, arXiv 0608360). In this paper, the authors formulate abstract criterions for a set X, its set ...
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Homotopy of parts of surface (?)

Given surfaces $A$ and $B$ as below : Note that $A$ is obtained by collapsing all boundary to a point (all purple lines and points to one point) and $B$ is obtained by collaping red part to one point,...
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Embeddings of a surface that are given by linear systems of divisors

I am reading the proof of Castelnuovo's contractibility criterion in Beauville's Complex Algebraic Surfaces. I would like to clarify a paragraph. We have a hypeplane section $H$ of a surface $S$, a ...
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Divergence theorem and test function

Consider Let $\Omega \subset \mathbb{R}^2$ denote the open unit ball in $\mathbb{R}^2$. The unbounded function $$u(x) = log log(1+\frac{1}{|x|})$$ is given. Consider the compactly supported ...
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Beauville Fact III.22

I had the following doubt regarding the proof of Fact III.22 in Beauville's book on algebraic surfaces: What I don't get is the second application of the projection formula. How are we applying the ...
I want to compute the surface of the solid given by $$(x^2+y^2+z^2)^2=z^2\,\,\,\,\,\,\,\,\,\,(1)$$ I thought about spherical coordinates, i.e.(x,y,z)=(r\cos\phi\sin\theta,r\sin\phi\sin\theta,r\cos\...
I want to compute the area of the boundary of the solid $x^2+z^2\leq 1$ and $y^2+z^2\leq 1$ to compute his surface afterwards. But I somehow don't see how. First I thought about polar coordinates, ...