For questions about surfaces.

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geodesic polar coordinate parallel circles

When is it possible to have the same constant geodesic curvature on all parallels of a constant Gauss curvature surface?
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67 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
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Revolution surfaces of constant Gaussian curvature k=1

Prove that all revolution surfaces $(\phi(v) \cos u ,\phi(v) \sin u,\psi(v)) $ of constant Gaussian curvature $k = 1$ is one of the following types: $\phi(v)=C\cosh v$ and $\psi(v)=\int_0^v ...
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2answers
41 views

Viviani on Sphere parametrization

How should parametrization of the 2 parameter surface of a sphere (latitude u, longitude v) be changed to result in 1 parameter curve of Viviani?
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1answer
26 views

How to calculate volume and surface area of three dimensional figures given set of three dimensional coordinates?

I have set of three dimensional coordinates, and the shape is unknown. I would like to calculate the surface area and volume for these coordinates approximately. What is the right approach to solve ...
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3answers
87 views

The principal curvatures of a surface of revolution

The principal curvatures of the surface at a point is defined as the maximal and the minimal curvature among all normal sections. It's claimed (say, on Stillwell's Geometry of Surfaces) that for a ...
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24 views

Two unit disks spliced together?

$$ x^2 + (y-z)^2 = 2 x^2 z/y $$ The surface represented by above equation is formed by radial cuts on two separate unit diameter disks spliced together forming a "continuous" surface around ...
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1answer
29 views

Finding the surface area of a parametrized surface

I was wondering how you would compute the surface area of a parameterized surface. Is there a formula or set of procedures you can follow to compute this. Say I wanted to compute the surface area of a ...
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1answer
22 views

Total variation as surface area smooth functions of two variables.

I learnt we have different definitions for the total variation for functions of the form $f:\mathbb{R}^2\to\mathbb{R}$ which are in some way analogous to the total variation of functions of one ...
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21 views

Orientation of surfaces

From the book: Fixing a parametrization $x(u,v)$ of a neighborhood of a point $p$ of a regular surface $S$, we determine an orientation of the tangent plane $T_p (S)$, namely, the orientation of the ...
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1answer
44 views

Inverse mapping for a simple $\mathbb{R}^3$ surface given by $(\sin u, \sin 2u, v)$.

For a domain $U=\{\, (u,v) \in \mathbb{R}^2 \mid -\pi<u<\pi,\ 0<v<1 \,\}$ we have a mapping $X \colon U \to \mathbb{R}^3$ defined by $X(u,v) = (\sin u, \sin 2u, v)$. The resulting surface ...
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23 views

“Circle” on pseudosphere

How should parametrization of the 2 parameter surface of a pseudosphere ("latitude" u, longitude v) change to result in a 1 parameter curve of constant geodesic curvature?
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2answers
30 views

Circumference of a circle [closed]

Three circles of radius r are circumscribed by a bigger circle , such that the centres of smaller circle form a triangle when joined . what is the circumference of the bigger circle
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1answer
48 views

Curvature proof of a convex plane curve

Having a little trouble with part b. Is there a way to show that this curve would be arc length paramaterized? I am assuming that we cannot say this. If it is not we can take alpha', alpha'' and ...
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1answer
56 views

The projections are differentiable $1$-forms.

Suppose $M$ is a surface and suppose $X: U \subset \mathbb R^2\rightarrow M$ is a coordinate patch. Then for every $p \in X(U)$, the pair of vectors $(X_u(X^{-1}(p),X_v(X^{-1}(p) )$ is a basis of the ...
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2answers
247 views

Hilbert theorem and constant negative curvature surfaces

Let us consider the tractroid (pseudosphere) obtained by rotation from the tractrix curve. The surface is not defined on the "big rim", so it is not a complete set. Hilbert's theorem states that there ...
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2answers
215 views

Complete developable surface in $\mathbb{R}^3$ is ruled

Let $X \subset \mathbb{R}^3$ be a complete smooth surface which is developable in the sense that its Gaussian curvature is identically zero. Wikipedia claims that such a surface is necessarily ruled, ...
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1answer
227 views

Classification of fundamental groups of non-orientable surfaces

I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$. I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to ...
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47 views

Prove that the line is tangent to the curve at the point.

Hello can someone please walk me through part a and b of the below question? I really want to understand it but am having a hard time figuring out the solution. I know how to calculate curvature for a ...
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2answers
99 views

Elliptic curves on a K3 surface

Let $X$ be an elliptic K3 surface. Let $\alpha$ be a smooth curve of genus $\geq3$. Define $$d(\alpha)=\min\lbrace \epsilon\cdot \alpha \ | \ \epsilon \mbox{ is an elliptic curve on } X \rbrace, $$ ...
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1answer
51 views

Give a closed plane curve C with k (curvature) > 0 that is not convex, Draw closed plane curves with rotation indices 0, 2, -2, and 3

1.) Give a closed plane curve C with k (curvature) > 0 that is not convex. can someone please explain these concepts to me. How can you have a closed plane curve like this? Do you used signed ...
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2answers
189 views

Self-intersection of parametric surface using Gauss-Bonnet theorem

I am trying to detect when a closed parametric surface intersect itself. My surface is described as a triplet of parametric functions $x(u,v)$, $y(u,v)$ and $z(u,v)$ where $u,v\in[0,1]$. For that ...
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23 views

sphere parametrization

We have the standard spherical surface parametrization in which one set describes geodesics (longitudes) and another ( latitudes/parallels). What parametrization may be possible for both sets to be ...
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0answers
34 views

A converse proof that involves Torsion, curvature, and differentiation that equates to 0

I am having difficulty proving the converse in part B. I understand part A and have shown that t/k-t/k = 0. I found that n=-1/k, n'-bt= b'+tn = 0, so n' = bt and b'=-tn. However, I am unable to find ...
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85 views

A curve internally tangent to a sphere of radius $R$ has curvature at least $1/R$ at the point of tangency

Suppose $a$ is an arc length-parametrized space curve with the property that $\|a(s)\| \leq \|a(s_0)\| = R$ for all $s$ sufficiently close to $s_0$. Prove that $k(s_0) \geq 1/R$. So, I was going ...
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39 views

surface presentation

Given the following group and presentation, how could I go about showing if there exists a compact surface with that fundamental group? The group is $\big \langle a, b, c, d, e $ $\mid$ ...
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36 views

What does this operator $\odot$ mean

I read this about the second fundamental form in Wikipedia and I’ve no idea what does $\odot$ mean? Does anybody know? $$II=-dN\cdot dP=\omega^3_1\odot\omega^1+\omega^3_2\odot\omega^2$$
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1answer
12 views

Equation for multiple peak surface to test particle swarm optimising algorithm

i have developed a particle swarm optimisation algorithm that i am running some tests on. It is able to solve simple equations like this: $x^2 + y^2 + 300y - 254x + 3$ with only one optimum but ...
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2answers
39 views

Describe the surface whose equation is given by $x^2+y^2+z^2-y=0$. [closed]

Describe the surface whose equation is given by $x^2+y^2+z^2-y=0.$ Show your working.
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3answers
51 views

Proof that this surface is of revolution

I have a surface with parametric equation $$\mathbf{x}(u,v)=(u\cos(v),u\sin(v),u^2),$$ $u$ is any real number, $v$ is between $0$ and $2\pi$. I don't know how to show that this is surface of ...
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1answer
254 views

Restriction of a differentiable map $R^3\rightarrow R^3$ to a regular surface is also differentiable.

This is again an excercise from Do Carmo's book. Prove: if $f:R^3 \rightarrow R^3$ is a linear map and $S \subset R^3$ is a regular surface invariant under $L,$ i.e, $L(S)\subset S$, then the ...
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1answer
34 views

Classical presentation of fundamental group of surface with boundary

It is well known fact about fundamental group of orientable compact surface: Letting $g$ be the genus and $b$ the number of boundary components of surface $M$. There is a generating set ...
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1answer
305 views

Find the surface area obtained by rotating $y=1+3x^2$ from $x=0$ to $x=2$ about the y-axis.

Find the surface area obtained by rotating $y= 1+3 x^2$ from $x=0$ to $x = 2$ about the $y$-axis. Having trouble evaluating the integral: Solved for $x$: $x=0, y=1$ $x=2, y=13$ $$\int_1^{13} ...
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40 views

Strain mapping sphere to plane

I need general indications or guidance. I do not know how to map a surface $ z = \sin (\pi x) \sin (\pi y), (x,0,\pi), (y,0,\pi)$ to a unit square. Nor do I know how to map a quadrant of a unit ...
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26 views

lifting loops on surface with abelian fundamental group for decrease their self-intersection number

I found a proof of following statement (which is available here ), but I'm not sure if we must assume that fundamental group of surface $M$ is non-abelian: Lemma: Let $M$ be a compact orientable ...
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1answer
57 views

Surface Integral over a sphere

Suppose $f(x,y,z)=g\left(\sqrt{x^2+y^2+z^2}\right)$, where $g$ is a function of one variable such that $g(2)=-5$. Evaluate $$\iint_S f ~dS,$$where $S$ is the sphere $x^2+y^2+z^2=4$. Now, I ...
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1answer
20 views

Number of intersections of two closed loops on a genus zero surface

I have stumbled onto the following fact and I am quite helpless in seeing why this is true (although I can agree intuitively). Let $M$ be a surface of genus zero (open or closed, with or without ...
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0answers
27 views

Finding the leftmost, rightmost, top, and bottom, points, on a surface, of a sphere.

So I'm making a 3D game, and the player is inside a glass sphere. I'm projecting a bunch of points onto the sphere, and I need to find the leftmost, rightmost, topmost, and bottommost points, so I can ...
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2answers
48 views

1-dimensional foliation on a surface

Is it possible to find a 1-dimensional nonsingular foliation on an orientable surface with one boundary component such that lines of the foliation are transverse to the boundary?
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1answer
39 views

Why does every noncompact orientable surface have a complex structure?

There is a high-powered proof of the fact that orientable noncompact surfaces have free fundamental group here that invokes the ability to put a complex structure on any such surface. But why should ...
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1answer
19 views

How do I find the ridges and valleys given a surface elevation function

Given a surface with a single elevation value for every x and y how can I find the places where the isoelevation contours have the tightest bends? And how can I differentiate between bends that are ...
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1answer
59 views

Is it true that $d\textbf{S} = dy dz\textbf{ i }+ dx dz\textbf{ j }+ dx dy\textbf{ k }$

I came up with this in my mind, Just wondering if it is true I am thinking about it too, will post my observations, if any
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1answer
57 views

Relationship between Surface Area and Volume

Question: Is there a general relationship between surface area and volume analogous to the below examples? Example 1. Consider a ball $B$ centered at the origin of a spherical coordinate system. The ...
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1answer
133 views

How to construct pseudospherical surfaces from sine-Gordon solutions?

Due to my not being very skilled in differential geometry, I want to ask if there is a reference (book, paper, etc.) that explicitly works out how one constructs the parametric equations of a ...
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1answer
199 views

Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf

Let $B$ a smooth projective connected variety over $\mathbf C$. Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero. Does the converse hold? That is, suppose that $B$ ...
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1answer
44 views

Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
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1answer
19 views

Check if a point is within a quadratic surface (with arbitrary rotation)

Is there a general way to check whether a point is on a quadratic surface given that the principal axes do not need to coincide with the coordinate axes and that the quadric's centroid does not need ...
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41 views

differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
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1answer
227 views

If radial projection is bijective then is it a homeomorphism?

Suppose $S$ is a regular surface in $\mathbb{R}^3 $ and $0\not\in S$. Now consider the radial projection $f: S\to\mathbb{S}^2$ given by $$f(x)=\frac{x}{||x||} \hspace{5mm}\mbox{ for all $x\in S$}$$ ...
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1answer
22 views

Optics surface equation to quadric form

This should be straightforward, but honestly I forgot even the names Google for... I've got a surface description in this form (what is it called?): $$z=\dfrac{cr^2}{1+\sqrt{1-(1+k)c^2r^2}},$$ ...