Surface is a two-dimensional submanifold of three-dimensional Euclidean space.

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Cartesian/Parametric 3d equation of a cheese twist?

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks e.g. ...
2
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1answer
42 views

Surface area of a slightly deformed sphere

Consider the unit sphere, which can either be described by $x^2+y^2+z^2=1$ or by the equation $r(\theta,\phi)=1$, where $(r,\theta,\phi)$ are spherical polar coordinates. I define a deformed sphere ...
0
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1answer
50 views

Volume calculating using double integral

Here is my task: Calculate the volume under the surface $z=x^{2}-y^{2}$ over the region $(x^{2}+y^{2})^{3}=a^{2}x^{2}y^{2}$. Before solving this task, let's say that $z=x^{2}+y^{2}$ instead ...
2
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1answer
24 views

Find all surfaces that can be obtained from an octagon by identifying edges in pairs.

Find all surfaces that can be obtained from an octagon by identifying edges in pairs. I think there are many many surfaces. Can anyone give some hints for the question?Thanks.
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1answer
13 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
1
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0answers
35 views

Difference between quadric and conic

What is the difference between a conic and a quadric? I'm guessing that this depends on your ambient space? I think that conics are just special quadrics and are a codimension 1 object and a quadric ...
2
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2answers
41 views

Geodesics on a generalized cylinder

I want to prove that given a generalized cylinder $C(s,t)=\alpha(s)+t\hat{z}$ , where $\alpha$ is a curve on the $xy$ plane and $\hat{z}$ is the $z$-axis vector, then a geodesic curve $\gamma$ has the ...
2
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0answers
64 views

Fundamental Group of a Surface [closed]

I came across the term "fundamental group of a surface" while reading a paper, and I'm not sure what it it all about. As well, what is understood by the generators of the fundamental group of a ...
0
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0answers
29 views

decomposition of a closed surface

I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g-1)$ pants bounded by 3 geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...
4
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2answers
571 views

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 ...
1
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1answer
55 views

Orientability and Hypersurfaces

I got stucked in this problem: Show that: i) Every embedded closed hypersurface $S$ is orientable. ii) Every differentiable hypersurface defined by a regular cartesian equation $\ g(x_1,..., x_n)=0$ ...
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0answers
19 views

Closest points on two skew surfaces [closed]

I am trying to find the two closest points on two surfaces. The surfaces are not extended infinitely but are restricted by some boundaries. Th possible set of surfaces could be combination of ...
1
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0answers
27 views

Change of variable and diffeomorphic surfaces?

Suppose two curves $\gamma$ and $\gamma'$ are diffeomorphic. Is the arc-length measure $ds_\gamma$ absolutely continuous to $ds_\gamma'$ with a positive derivative? ($ds_\gamma=\phi\, ds_\gamma'$ for ...
2
votes
0answers
31 views

Homeo- and diffeomorphism groups of oriented surfaces

I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group ...
1
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1answer
24 views

The set of fixed points of a $C^1$ involution is a surface.

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a $C^1$ involution i.e $f\circ f=id$. Show that the set $Fix(f)=\{x \in \mathbb{R}^n;f(x)=x\}$ of fixed points of $f$ is a surface of $\mathbb{R}^n$ ...
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2answers
58 views

Polygonal presentations: why no two-letter words?

In Lee's book Introduction to Topological Manifolds, he discusses polygonal presentations of surfaces. He does so by means of words $W_1, \dotsc, W_n$ such that each letter that appears must appear ...
1
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1answer
31 views

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$

Show by definition that $M=\{(x,y,z)|36x^2+4y^2-9z^2=36\}$ is a surface in $\Bbb R^3$. Definition A surface in $\Bbb R^3$ is a subset $M$ of $R^3$ such that for each point $p$ of $M$ there exists a ...
2
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1answer
41 views

Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
0
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0answers
19 views

Hyperbolic length does not depend on the subdivision.

I'm reading some notes on hyperbolic surfaces by François Labourie and there's an exercise I can't figure out. I have to prove that the length l(c) of a curve does not depend on the subdivision. It's ...
4
votes
1answer
68 views

Fundamental polygon square $abab$

What is the most convenient description of the space with fundamental polygon a square, with all vertices identified, glued by $abab$? If we were to identify only opposite vertices, we would get ...
0
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1answer
23 views

Parameterizing part of sphere

the part of the sphere given by: $$ S = \{ (x,y,z) | x^2+y^2+z^2 = 25, -4 \leq x,y,z \leq 4 \} $$ first Q: I'm not sure if I can apply to this Divergence theorem ? It seem that in order to use it I ...
2
votes
2answers
52 views

Proving that every patch in a surface $M$ in $R^3$ is proper.

Problem Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image ...
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21 views

Surface mentioned by Sophie Germain

Sophie_Germain_Surface Can someone please help going through the original paper no.22 in French? Does the doubly curved surface mentioned by Sophie Germain (Wiki) $$ 4 \frac{z^p-1}{z-1} = y^2 \pm ...
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1answer
82 views

About timelike surfaces with non-diagonalizable shape operator.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, with $${\rm d}s^2 = {\rm d}x^2+{\rm d}y^2 - {\rm d}z^2.$$ Take a differentiable surface $M \subset \Bbb L^3$, ...
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votes
2answers
20 views

How to simplify this equation with change of variables,

I have the equation, after completing the square: $$(x+\frac{y}{2})^2 + \frac {3y^2}{4} + z^2 = 1$$ How can I further simplify this equation? I need to find the volume inside of this surface. ...
0
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2answers
33 views

Find the volume enclosed by the surface $S := \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}$

Find the volume enclosed by the surface $$S := \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}.$$ My attempt was this: I moved the tricky $xy$-term over to the r.h.s. I now have $$x^2+y^2+z^2 = 1-xy,$$ ...
1
vote
1answer
49 views

Computing the volume inside a surface S, using a seemingly unrelated result,

Consider the surface $$S = \{(x,y,z): x^2 + xy + y^2 + z^2 = 1\}$$. What is the volume inside S? This is actually part (b) of the question. I'm not sure which approach to take. But part (a) of the ...
0
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0answers
17 views

Rotation Invariant Descriptors for Bivariate Polynomial Surfaces

I start with a simple example. Consider: $$ z = x^2 + y $$ and $$ z = y^2 + x $$ Visually speaking, both of these are essentially the same surfaces rotated by 90 degrees about the z-axis. I am ...
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0answers
41 views

Complements of homeomorphic subsets of surfaces homeomorphic?

Let us consider a surface $S$ and a subset $T\subset S$, where $T\cong D^2$ are homeomorphic and $D^2$ is an open disc. Let $T\cong Q\subset S$ be homeomorphic. Are $S\setminus T $ and $S\setminus Q$ ...
3
votes
2answers
41 views

Notion of curvature for a volume embedded in $R^3$

This question might sound slightly vague, but please bear with me. If I have an orientable, closed, sufficiently smooth surface in $R^3$, I can define its principal curvatures, mean curvature as ...
1
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1answer
15 views

How to graph the intersection of two surfaces

Another way of asking this is: How would one graph a 1 dimensional line (the intersection) in a 3 dimensional space. Some context for the question: Let's say I have two spheres ...
2
votes
1answer
27 views

An inequality for absolute total curvature in Riemannian surfaces

Let be $M\subseteq \mathbb{R}^3$ a compact (Riemannian) surface and let be $K$ the gaussian curvature of $M$. I want to prove that $$ \int_{M} |K| \geq 4\pi(1+g(M))$$ where $g(M)$ is the genus of ...
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0answers
11 views

Is there a pluricanonical divisor on a relatively minimal complex elliptic surface that can be written as sum of fibres?

A complex algebraic surface $S$ is said to be elliptic if there are a smooth curve $B$ and a surjective morphism $p \colon S \to B$ whose generic fibre is an elliptic curve (i.e. a smooth curve of ...
2
votes
1answer
27 views

Choose the reflection planes of a surface through a single point.

Let $S$ be a surface in $R^3$, for which coordinate vector field of $S$ has zero mean on $S$. Assume that for any vector $n$, a normal plane to $n$ exist, such that $S$ is symmetric about it. How can ...
10
votes
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112 views

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 ...
0
votes
1answer
22 views

How to work out the angle of a line passing through a plane

I have a triangular plane composed of three points. From this it it easy to deduce that the plane is in fact composed of two vectors which must touch at some point. because all of this is relative, ...
5
votes
2answers
69 views

Does intrinsic mean existing regardless of some bigger space?

How is the arc-length of a regular parametrized curve in a surface $S\subset\mathbb{R}^3$ intrinsic? Let $\bf{x}\rm(u,v)$ be a parametrization of $S$. Letting $E,F,G$ denote the coefficients of the ...
0
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1answer
19 views

Question about determinig types of surfaces?

$$x^2 +y^2 +z^2 +2x +1=0$$ This is an equation for dot if we are talking about surfaces, right? It is not an ellipsoid.
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votes
2answers
23 views

Find the volume of the region by integration

I have to find the volume of the region bounded by $x + z = 1$;$ y + 2z = 2$;$ x = 0$;$ y = 0 $;$ z = 0$; I tried to sketch the graph separately in the $y-z$ plane and then in $x-z$ plane. But I am ...
0
votes
3answers
47 views

What substitution can be used to evaluate the integral giving the area of the surface of revolution of the curve $x = \sqrt[3]{y}$?

The question is: Find the area of the surface generated by revolving the given curve about the $x$-axis: $$x = \sqrt[3]{y}, \qquad 1 \leq y \leq 8.$$ Now, all is well, simple enough question, ...
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22 views

Prove for criterion that two curve families are orthogonal on a surface in 3D

Let $E, F, G$ be the coefficients of the first fundamental form of a regular surface $R = R(u, v).$ Let $f(u, v) = c$ and $g(u, v) = d$ be two families of regular curves defined in the ...
4
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1answer
56 views

When does $\pi_1(\Sigma)$ inject into $\pi_1(S^3 \setminus \Sigma)$?

Here's a fun fact from knot theory: $\quad$ If $\, \Sigma$ is a minimal-genus Seifert surface for a knot $K$, then $i_*:\pi_1(S^3 \setminus \Sigma) \to \pi_1(S^3 \setminus K)$ is injective, where ...
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0answers
26 views

what would the equation of a torus be by making the circunference $(y-2)^2+ z^2 = 0$ and $x=0$ turn along the $z$ axis

What I understand of the question is that I have to, somehow, give the equation of the torus that results of spinning the circumference $$(y-2)^2 + z^2 = 0$$ and $$x=0$$ which as far as I know is just ...
0
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2answers
17 views

Formula to find lenght of material I need to make a sprint

could somebody write me if you know how to calculate the lenght of a material i need to make a certain spring. I need any true formula you have.
1
vote
1answer
17 views

Parameterize the following surface: $x +y +z=1$, $x,y,z>0$.

I need to parameterize the following surface: $x +y +z = 1$, $x,y,z>0$. I tried to put: $\sigma(u,v)=(u+v,u-v,-2u+1)$, but does it solve the case of $x,y,z>0$?
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1answer
22 views

Ruled surface out of lines of curvatures

I'm trying to proof the following statement: Let $c$ be a curve inside a surface element $f:U\rightarrow\mathbb{R}^3$ (i.e $c=f\circ\gamma$ where $\gamma:I\rightarrow U$). Then $c$ is a line of ...
2
votes
1answer
53 views

Need help understanding a relation between the fundamental forms

The book I am reading briefly mentions this relation between the fundamental forms but gives no explanation of how they got it. Take the following as the Weingarten Map/Shape Operator where $\nu$ is ...
2
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0answers
24 views

Gradient of second fundamental form

In the book I'm reading ("Differential Geometry Curves-Surfaces-Manifolds by Wolfgang Kuhnel") two definitions of prinicpal curvatures directions are presented: The extramum values of $II(X,X)$ ...
3
votes
3answers
174 views

What surface is represented by the following equation

$$\sqrt[3]{x^{2}}+\sqrt[3]{y^{2}}+\sqrt[3]{z^{2}}=1$$ Taking cubes of both sides only leads to a more complicated formula. How should one interpret this one. And, also if you could point me to a tool ...
2
votes
2answers
67 views

Differential of a rotated f(x, y) surface

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial ...