Parametrization of a Cone such that $E=G$ and $F=0$

Question

How would we parametrize a regular conical surface such that $\dfrac{x^2+y^2}{c^2} = z^2$ to have the first fundamental form $E=G$ and $F=0$?

I'm asking this so that we can ensure the existence of a conformal map.

Thanks

Answer 1

Hint: Try to find a surface of revolution parametrization of the form

$$ X(\rho, \theta) = \left( h(\rho) \cos \theta, h(\rho) \sin \theta, \frac{h(\rho)}{c} \right). $$

We have $$ X_{\rho} = \left( h'(\rho) \cos \theta, h'(\rho) \sin \theta, \frac{h'(\rho)}{c} \right), \\ X_{\theta} = \left( -h(\rho) \sin \theta, h(\rho) \cos \theta, 0 \right). $$

Then $$ E = \left< X_{\rho}, X_{\rho} \right> = \frac{1 + c^2}{c^2} h'(\rho)^2, \\ F = \left< X_{\rho}, X_{\theta} \right> = 0, \\ G = \left< X_{\theta}, X_{\theta} \right> = h(\rho)^2. $$

The condition $E = G$ gives you a differential equation on $h$ which you can solve to obtain the required parametrization.