Classify the surface problem?

G

Guest

Guest
Hi
I'm having trouble with the consistency on how to classify surfaces.

Here is the problem:
The equation p^2cos(2phi) = 1 is the equation of a quadratic surface expressed in spherical coordinates. Write this equation in rectangular coordinates and classify the surface.

Here are all the different equations to use:
x=psin(phi)cos(theta); y = psin(phi)sin(theta); z = pcos(phi)
p^2 = x^2 + y^2 + z^2

Thanks for any help you can give me on this. :)

Take care,
Beckie
 
p^2(cos(2phi))=1...

To classify it, I'd put it into terms of x,y, and z first, since they're more familiar to deal with.

Well, I'd start off with using the double-angle formula for cosine :
cos(2x)=(cosx)^2-(sinx)^2

Applying that, and distributing the p^2, we get
p^2(cos(phi))^2 - p^2(sin(phi))^2 = 1

(p cos(phi))^2 - (p sin(phi))^2 = 1

p cos(phi) = z
p sin(phi) = r

So we've got

z^2-r^2=1

r^2=x^2+y^2

So, the final equation is z^2-x^2-y^2=1.

which is in the form -(x^2/a^2)-(y^2/b^2)+(z^2/c^2)=1, which corresponds to a hyperboloid of two sheets.
 
Thank you thank you!!! Is there a way to plot points of these equations so I can graph it? I need to have a visual of how this works.

Thanks so much
Take care,
Beckie
 
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