# Find a parametric representation of a cone

• Oct 27th 2009, 05:53 PM
Korgoth28
Find a parametric representation of a cone
Find a parametric representation of the cone: $\displaystyle z=\sqrt{3x^2 + 3y^2}$ in terms of the parameters $\displaystyle \rho$ and $\displaystyle \theta$ where $\displaystyle \rho$, $\displaystyle \theta$, and $\displaystyle \Phi$ 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 $\displaystyle \rho$s in particular. Thanks in advance!
• Oct 28th 2009, 05:22 AM
HallsofIvy
Quote:

Originally Posted by Korgoth28
Find a parametric representation of the cone: $\displaystyle z=\sqrt{3x^2 + 3y^2}$ in terms of the parameters $\displaystyle \rho$ and $\displaystyle \theta$ where $\displaystyle \rho$, $\displaystyle \theta$, and $\displaystyle \Phi$ 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 $\displaystyle \rho$s in particular. Thanks in advance!

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

If we set y= 0 so we are looking at the x,z-plane, the equation becomes $\displaystyle z= \sqrt{3}x$ and, since that has a constant slope, $\displaystyle cot(\phi)= \sqrt{3}$ so $\displaystyle \phi= \frac{\pi}{6}$. Then $\displaystyle sin(\phi)= \frac{\sqrt{3}}{2}$ and $\displaystyle cos(\phi)= \frac{1}{2}$. Put those into the equations for x, y, and z in spherical coordinates.
• Oct 28th 2009, 08:17 AM
Korgoth28
Quote:

I wonder if you weren't mistakenly thinking you were supposed to have http://www.mathhelpforum.com/math-he...c8912759-1.gif and http://www.mathhelpforum.com/math-he...2cc525ec-1.gif as parameters?
That was exactly my problem, I just didn't read it carefully enough. Thanks so much!