All Questions
Tagged with solid-of-revolution differential-geometry
17
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Generalization of why the area of a surface of revolution is not $2 \pi \int_{a}^{b} (y) dx$
Based on the posts Areas versus volumes of revolution: why does the area require approximation by a cone? and Why is surface area not simply..., approximating an ND quantity in an ND region using N-1D ...
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surface $X$ is a surface of revolution if and only if there is a line $L$ such that all planes through $L$ meet $X$ in geodesics.
I try to solve Exercise $122$ on page 40 of this pdf
Show that the surface $X$ is a surface of revolution if and only if there is a line $L$ such that all planes through $L$ meet $X$ in geodesics.
...
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Prove that the parametric surface of revolution has continuous inverse of its image
Let $\boldsymbol{\sigma}(u,v)=\big(f(v)\cos u, f(v) \sin u, g(v)\big)$, where
$U=[(u,v):0<u<2\pi,~v\in I]$ ($I$ open interval of $\mathbb{R}$) be the parametric surface of revolution by rotating ...
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2
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Obtaining the Surface Area of a Superegg with a Given Volume
I have been stuck trying to find an expression for the surface area of a superegg of a given volume. Specifically, the shape I'm looking at is the solid of revolution obtained by rotating a squircle (...
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Local isometry between half a disk and the cone of revolution $3(x^2+y^2)=z^2$
This is an exercise from my Differential Geometry course:
Define the function $\Phi:\left]0,2\right[\times \left]-\pi/2,\pi/2\right[\longrightarrow \mathbb{R}^2$, $\Phi(\rho,\theta)=(\rho\cos \theta,\...
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Surface of revolution of $z=y^4$
I've read in a book that the surface of revolution of the curve $y=z^4$ around the $z$ axis is an example of a regular surface which has a plane point (in the origin), but it is completely contained ...
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What is the gradient of $f (x, (y^2+z^2)^{1/2})$?
Consider a $c \in \mathbb R$ and a function $f: U \subset \mathbb R^2\rightarrow \mathbb R $ with $ \nabla f (p) \neq 0, \forall p \in f^{-1}(c)$, where $U$ is contained in the upper half plane y >...
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Equal area parameterization of a torus?
I am trying to parameterise a surface of revolution such that each infinitesimal area element is uniform across the surface. The cross-sections of the surface are shown in the picture below. The title ...
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Elipse to a spheroid? [closed]
I'm having a problem trying to prove that an elipse, given by
$$ \xi = \left\{(x,y,z)\in \mathbb{R}^3 : y = 0, \frac{x^2}{a^2}+\frac{z^2}{c^2} = 1\right\} $$
if we rotate to the z-axis is a spheroid ...
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Distance between two points in a solid of revolution
Let $f: I \to (0,\infty)$ be a smooth and positive function. Let
$$\Sigma_f = \{(x,f(x)\cos\theta,f(x)\sin\theta) : x \in I, \theta \in \Bbb{R}\}$$
be a solid of revolution generated by rotating ...
- 422
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Show surface of revolution is an orientable surface
If I have a curve parametrized in the form $(f(v),0,g(v)),f>0, v\in(a,b)$, rotating this curve about the $z$-axis. I get a surface of revolution $S$ and I know it is a regular surface as it can be ...
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Parallel transport on a surface of revolution along parallels and meridians
I want to find the parallel transport on a surface of revolution along parallels and meridians. I considered the following approach:
take the parametrization of a surface of revolution
$$
\mathbf{r}(...
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Implicit formula for surface of revolution
I want to find the unit normal for a surface of revolution of the form
$F(t,s)=(r(t)cos(s),r(t)sin(s),z(t))$
where $\gamma(t)=(r(t),z(t))$ is a curve with unit speed and $r(t)>0$.
I know that if ...
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Willmore energy of revolution torus
I would like to know if it is possible to prove the Willmore Conjecture over the particular case of a revolution torus having the knowledge of a first course in differential geometry.
Suppose $0 < ...
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Surfaces of revolution with curvature 0
I am trying to find all the surfaces of revolution with Gaussian curvature $K \equiv 0$.
This is what I got so far. If we assume the surface of revolution is parametrized by $(\varphi(v) \cos u, \...
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