Change in the topology of $f^{-1}(c)$ as $c$ passes through the critical value.

Question

Pictorially examine the catastrophic change in the topology of $f^{-1}(c)$ as $c$ passes through the critical value, where $$f (x, y, z) = x^2 + y^2 - z^2.$$

I don't have the slightest idea about how to do this. I got that $$df_v = (2x, 2y, -2z)|_v.$$

Any idea? Thanks.