Converting ρ(1-2cos²φ)=-ρsin²φ to cylindrical and rectangular

[DeadxBunny]

Question:

(Note: p=rho and o=phi)
Convert p(1-2cos^2(o))=-psin^2(o) into cylindrical and rectangular coordinates and describe or sketch the surface.

The part that I don't know how to do is converting the spherical equation into cylindrical or rectangular coordinates. I know all the equations like x=psin(o)cos(theta) and y=psin(o)sin(theta) but I don't see how I can manipulate the given equation so that I could use those equations. Any help would be greatly appreciated!

 

[cepheid]

You posted in the Calculus section as well. See response there. (just an FYI Cross posting is generally frowned upon...don't want someone to yell at you).

 

[wais]

To convert this spherical equation into cylindrical coordinates, we can use the following relationships:

p = r
o = theta

Substituting these values into the given equation, we get:

r(1-2cos^2(theta)) = -rsin^2(theta)

Next, we can use the relationships between cylindrical and rectangular coordinates:

x = rcos(theta)
y = rsin(theta)

Substituting these values into the equation, we get:

x(1-2y^2) = -y^2

This is the equation in rectangular coordinates. To sketch the surface, we can rearrange the equation to get y as the subject:

y = ±√(x/(2x+1))

This is a hyperbolic curve in the xy-plane. The surface created by this equation would be a cone with its vertex at the origin, extending infinitely in both directions. The base of the cone would be a hyperbolic curve in the xy-plane.