# Parametric surfaces

• Jun 24th 2008, 10:21 AM
Mathstud28
Parametric surfaces
My book skipped this section, but I think it is important, and I am having difficulty finding resources online. So if this is incorrect bare with me.

Say we have a curve defined by

$\displaystyle z^2=x^2+y^2$

Now we want to parametrize this, but we do not want z's parameter to be dependent on x and y's. So is this how you would paremetrize it?

Let $\displaystyle z=v$

and $\displaystyle x=v\cos(t)$

and $\displaystyle y=v\sin(t)$

I chose this because

$\displaystyle x^2+y^2=v^2\cos^2(t)+v^2\sin^2(t)=v^2=z^2$

Is this correct?

And secondly how do we integrate restrictions into this, for example

$\displaystyle z=x^2+y^2$

I believe that what I would do is let

$\displaystyle z=v$

and

$\displaystyle x=\sqrt{v}\cos(t)$

and

$\displaystyle y=\sqrt{v}\sin(t)$

So we would get

$\displaystyle x^2+y^2=v\cos^2(t)+v\sin^2(t)=v=z$

But what if the quetion also stated that it is the portion of this curve below $\displaystyle z=4$?

If someone could tell me if I am doing this right, and then maybe give me a hint for how to do the last part, that would be great.

Mathstud
• Jun 24th 2008, 10:25 AM
Moo
Hello,

Quote:

Originally Posted by Mathstud28
My book skipped this section, but I think it is important, and I am having difficulty finding resources online. So if this is incorrect bare with me.

Say we have a curve defined by

$\displaystyle z^2=x^2+y^2$

Now we want to parametrize this, but we do not want z's parameter to be dependent on x and y's. So is this how you would paremetrize it?

Let $\displaystyle z=v$

and $\displaystyle x=v\cos(t)$

and $\displaystyle y=v\sin(t)$

...

Uuuh... If you define z=v and x=v... and y=v... wouldn't z be dependent on x & y ? :confused:

This would help ya, depending on what exactly you're looking for : Cylindrical coordinate system - Wikipedia, the free encyclopedia

$\displaystyle x=r \cos(t)$

$\displaystyle y=r \sin(t)$

$\displaystyle z=z$

huh ? This is weird, the English version of Wiki isn't the same as the French one... Coordonnées cylindriques - Wikipédia (French)
• Jun 24th 2008, 10:32 AM
Mathstud28
Quote:

Originally Posted by Moo
Hello,

Uuuh... If you define z=v and x=v... and y=v... wouldn't z be dependent on x & y ? :confused:

This would help ya, depending on what exactly you're looking for : Cylindrical coordinate system - Wikipedia, the free encyclopedia

$\displaystyle x=r \cos(t)$

$\displaystyle y=r \sin(t)$

$\displaystyle z=z$

huh ? This is weird, the English version of Wiki isn't the same as the French one... Coordonnées cylindriques - Wikipédia (French)

Yeah, its hard to find on the internet, but kind of what I was saying is if we have that

$\displaystyle x=x(t)$
$\displaystyle y=y(t)$
and
$\displaystyle z=z(t)$

We would not be describing the whole surface, to describe any point on the surface we must let the last part be any real number, therefore it would be delegated its own parameter.