Questions tagged [surfaces]

For questions about two-dimensional manifolds.

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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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1answer
51 views

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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how can calculate gradiant of curvature of surface

I am solving question 7.2 Zangwill electrodynamics and I need to prove that $$ \vec{\nabla \Psi} = \partial\Psi/\partial n \enspace \hat{n} + \partial\Psi/\partial\tau_{1} \enspace \hat{\tau_{2}} ...
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56 views

homeomorphism $φ$ of $S_5$ of order $5$.

let $S_g$ be the closed, compact, orientable surface of genus $g$ and two homeomorphisms of $S_g$ are equal if they are isotop . we can find a homeomorphism $φ$ of $S_5$ of order $5$. consider ...
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orientation of a curve

In the eponymous wikipedia article we read the following definition of curve orientation: In the case of a planar simple closed curve (that is, a curve in the plane whose starting point is also the ...
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Question regarding planar and non-planar end of a surface

I am reading the paper On the Classification of Non-Compact Surfaces by Ian Richards. I have some questions about the statement of Theorem 3 on page 268. $\textbf{Theorem 3.}$ Every surface is ...
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Calculating surface of ellipsoid visible from any point outside of it

I have recently become interested in calculating the surface of Earth visible from any point in space. In previous questions such as Visible Portion of the Earth's Surface here and What is the ...
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1answer
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Curvature and geodesic of $M = \mathbb{R}\times \mathbb{R}_{>0}$ with metric $ds^2 = dx^2 + ydy^2$

Let $M = \mathbb{R}\times \mathbb{R}_{>0}$ with the metric $ds^2 = dx^2 + ydy^2$ a. Calculate the Gaussian curvature of the abstract surface $M$ b. Determine all geodesics in $M$ going through $(...
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Spine of a noncompact surface with proper retraction

I am reading this mathoverflow thread about the fact that "the fundamental group of a connected non-compact surface is free". One of the approaches is to first show every connected non-...
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why Alexander method gives us a finite combinatorial problem?

The Alexander Method in Farb and Maraglit's "A Primer on Mapping Class Groups" For a surface $S$ with marked points, we say that a collection $\left\{\gamma_{i}\right\}$ of curves and arcs ...
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If $M \subset \mathbb{R}^3$ is connected and every geodesic is planar, then $M$ is contained in a plane or a sphere

I have seen this question twice on MSE (here and here), but both questions were considered unanswered by the authors, and the hints there didn't really help me. So I decided to write a new inquiry ...
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1answer
68 views

What are the differences between hypersurfaces and manifolds?

I understand this question is just about the definitions, but I want to learn the general concepts in the mathematical community. As far as I know, a hypersurface is a certain type of manifolds that ...
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1answer
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Reference Request on a Necessary and Sufficient Condition for Isothermality

I am getting acquainted with the fundamentals of differential geometry for the sake of a problem I have been thinking about. In Green, G. M. "Some Geometric Characterization of Isothermal Nets ...
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1answer
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Approximate equation of a surface/curves to be used as an objective function

I would like to approximate the following equation to be used as an objective function in gurobi Multi-objective optimization. The continuous decision variables in the above equation are $ I_{i,v}$ ...
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Reason for a surface to be minimal

Let $B$ and $S$ be smooth irrational curves, and $G$ a group acting faithfully on $B$ and $S$, such that $B/G$ is elliptic and $F/G$ is rational. Why is that true that $S=(B\times F)/G$ is minimal? ...
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How to prove r = n * 2(l·n) - l in specular reflection?

I asked this question where I understand all basics concepts of specular reflection. From that question I read the reflection of the vector (r) across a normal...
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Gauss theorem applied for curl integral over surface of torus giving wrong answer

Two surfaces are given in cylinder coordinates as respectively $z=r-1$ (cone) and $ z^2 + (r-2)^2=1$ (torus, doughnut). Both surfaces are rotationally symmetric about x-axis. Define a vector field ...
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The definition of special cubic fourfolds

According to the basic paper "Special Cubic Fourfolds" (https://www.math.brown.edu/bhassett/papers/cubics/cubiclong.pdf, [BH98]) by Brendan Hasset, a special cubic fourfold is defined as ...
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Do carmo: theorem of turning tangents --- notational confusion

Theorem statement Let $\mathbf x: U \subseteq \mathbb R^2 \to S \subseteq \mathbb R^3$ be a parametrization compatible with the orientation of $S$. Assume further that $U$ is homeomorphic to the open ...
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what is definition of height function for subsurface?

what is definition of height function for subsurface ? in the paper : " On the TeichmuÈ ller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps.&...
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The classification of developable surfaces: Are these statements equivalent?

I thought to know very well the answer to the classification problem for developable surfaces, so I sought for some confirmation in the literature. However, what I encountered are seemingly ...
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Virtual cohomological dimension of mapping class group and braid group of punctured surfaces

Let $B_{k}(S_{g}),$ $MCG(S_{g};k)$ and $MCG(S_{g}))$ are Braid group, Mapping class group (relative to $k$) and Mapping class group of orientable surface $S_{g}$, respectively. For $g\geq3,$ we have ...
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Parametric Volume, volume analogue of parametric surface?

Is it possible to have a parametric volume? Is it possible to have a parametric volume in $\mathbb{R}^3$, $V:\mathbb{R}^3\rightarrow\mathbb{R}^3$, where $V$ is some function $V(x,y,z) = A(x,y,z)\hat{i}...
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What can we get via taking quotient of $\mathbb{C}P^1$ by a finite abelian group?

Let $G_n = \langle\tau\rangle$ and $G_m = \langle\sigma\rangle$ be groups of $n$-th and $m$-th roots of unity. Define action of $G_n \times G_m$ on $\mathbb{C}P^1$ as follows. $$ (\tau^k, \sigma^t) \...
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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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35 views

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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1answer
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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 ...
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1answer
35 views

How do I compute the surface of this solid?

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\...
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How do I compute the area of Steinmetz solid?

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, ...

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