Find a parametric representation of a cone

**Korgoth28** · October 27th 2009, 05:53 PM

Find a parametric representation of the cone: $z=\sqrt{3x^2 + 3y^2}$ in terms of the parameters $\rho$ and $\theta$ where $\rho$ , $\theta$ , and $\Phi$ are spherical coordinates of a point on the surface. I know the conversions for rectangular coordinates to spherical coordinates and vice versa, but for some reason I'm blanking out on how to do this, and how to get rid of the $\rho$ s in particular. Thanks in advance!

**HallsofIvy** · October 28th 2009, 05:22 AM

Originally Posted by Korgoth28

Find a parametric representation of the cone: $z=\sqrt{3x^2 + 3y^2}$ in terms of the parameters $\rho$ and $\theta$ where $\rho$ , $\theta$ , and $\Phi$ are spherical coordinates of a point on the surface. I know the conversions for rectangular coordinates to spherical coordinates and vice versa, but for some reason I'm blanking out on how to do this, and how to get rid of the $\rho$ s in particular. Thanks in advance!

Why to you want get rid of $\rho$ ? That is supposed to be one of the parameters. The point of using $\rho$ and $\theta$ as parameters instead of, say, $\theta$ and $\phi$ , is that $\phi$ is constant here. I wonder if you weren't mistakenly thinking you were supposed to have $\theta$ and $\phi$ as parameters?

If we set y= 0 so we are looking at the x,z-plane, the equation becomes $z= \sqrt{3}x$ and, since that has a constant slope, $cot(\phi)= \sqrt{3}$ so $\phi= \frac{\pi}{6}$ . Then $sin(\phi)= \frac{\sqrt{3}}{2}$ and $cos(\phi)= \frac{1}{2}$ . Put those into the equations for x, y, and z in spherical coordinates.

**Korgoth28** · October 28th 2009, 08:17 AM

I wonder if you weren't mistakenly thinking you were supposed to have

and

as parameters?

That was exactly my problem, I just didn't read it carefully enough. Thanks so much!

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