catenoid

[logistic_guy]

here is the question

Parametrize the surface (catenoid) that is formed when rotating the curve $\displaystyle x = \cosh z$ around the $\displaystyle z$-axis.


my attemb
i think they want me to find the map $\displaystyle \boldsymbol{\rho}: C \to \mathbb{R}^3$, where $\displaystyle C \subset \mathbb{R}^2$
i'm very good in differential geometry but it's shocking i can't visualize this curve. the variables $\displaystyle x$ and $\displaystyle z$ is confusing written together. i learn years ago in calculus solid of revolution and the function $\displaystyle y = \cosh x$ is easy to visualize. when i rotate this function around $\displaystyle y$-axis i get volume, its surface called catenoid. so i think this related to the original question but i can't gofrom $\displaystyle y = \cosh x$ to $\displaystyle x = \cosh z$ to the parametrize

 

[fresh_42]

WA gives you a nice picture




and a parameterization

$$\begin{array}{lll} x(u, v) &= \cos(u) \cosh(v) \\[6pt] y(u, v) &= \sin(u) \cosh(v) \\[6pt] z(u, v) &= v \end{array}$$
Maybe you need to shuffle the coordinates, your question has flaws. $(\sin,\cos)$ is the circle, $\cosh$ the curve.

 

[logistic_guy]

WA gives you a nice picture


View attachment 38909

and a parameterization

$$\begin{array}{lll} x(u, v) &= \cos(u) \cosh(v) \\[6pt] y(u, v) &= \sin(u) \cosh(v) \\[6pt] z(u, v) &= v \end{array}$$
Maybe you need to shuffle the coordinates, your question has flaws. $(\sin,\cos)$ is the circle, $\cosh$ the curve.

thank fresh_42 very much

i can't visiualize this drawing with $\displaystyle x$ and $\displaystyle z$. the parametrize become easy once you see the drawing in $\displaystyle x, y, z$ coordinate

the parametrize look simple after i see it. can you arrive to the same solution without WA or is it difficult?

 

[fresh_42]

You can directly write down the parameterization. We have a circle $(y(t),x(t))= (r\sin(t)\ ,\ r\cos(t))$ from the rotation, and the radius is determined by a corresponding function value $r=\cosh(s).$ The rotation axis is fixed and only determines where we are for the function value, thus $z=s.$

 

[logistic_guy]

You can directly write down the parameterization. We have a circle $(y(t),x(t))= (r\sin(t)\ ,\ r\cos(t))$ from the rotation, and the radius is determined by a corresponding function value $r=\cosh(s).$ The rotation axis is fixed and only determines where we are for the function value, thus $z=s.$

i'm already tell you once we see the drawing in $\displaystyle x,y,z$ coordinate the parametrize become easy. everything else is simple

do you see the parametrize after WA drawing or before WA drawing. imagine you're in a test where WA isn't your neighbor

 

[khansaheb]

Can you plot x = cosh(z) in x-z plane ? If you can, please show us the plot. You should NOT need WA for this!!

 

[fresh_42]

i'm already tell you once we see the drawing in $\displaystyle x,y,z$ coordinate the parametrize become easy. everything else is simple

do you see the parametrize after WA drawing or before WA drawing. imagine you're in a test where WA isn't your neighbor

I only used WA to see the picture. The parameterization can be done the way I wrote in post #4 without WA. Take a point $(z,y)=(z,\cosh(z))$ and a circle $(x(t),y(t))=(r\cdot\cos(t),r\cdot\sin(t)) .$ The points on the radius are at a distance $\cosh(z)$ from the $z$-axis. Now substitute $r$ by $\cosh(z).$ That's all.

Now you can rename the parameters - here $t,z$ - in $u,v$ or $s,t$ or whatever.

Your description in post #1 was confusing since you wrote $x=\cosh(z)$ and $y=\cosh(x).$ It is inevitable that you avoid such free use of variable names. You can name the objects whatever you want, but you must stick with it, or explicitly tell if you rename them.

What also becomes obvious in my opinion is that you have trouble imagining what is what. You should draw a lot more (with a pencil, not a pen) and doodle the situation on paper until a) you understand your own question, and b) be clear about what exactly are your difficulties before posting here. It's a bit like you post a brainstorming and leave it to us to make sense of it. Answers often just drop out if you are forced to precisely phrase the question. The more you spend on preparations, the more you can learn from the answers.

 

[logistic_guy]

Your description in post #1 was confusing since you wrote $x=\cosh(z)$ and $y=\cosh(x).$

I'm sorry I did that