 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2017/4/c/solution

Writing the coordinates as

(x, y, z)

, the level equation is

x^{2} + y^{2} - z^{2} = α

. Therefore:

$α = 0$ : a circular double cone, $x^{2} + y^{2} = z^{2}$ . Its vertex at the origin is the sole singular point; every other point is a smooth surface point.
$α = 1$ : a one-sheeted hyperboloid, with horizontal radius $1 + z^{2}$ . It is smooth everywhere.
$α = - 1$ : a two-sheeted hyperboloid, $z = \pm 1 + x^{2} + y^{2}$ . Both sheets are smooth everywhere, including their tips $(0, 0, \pm 1)$ .

The cone vertex is an instance of the double cone singularity. The tips of the two-sheeted hyperboloid are regular because the gradient

(2 x, 2 y, - 2 z)

is nonzero there.

 New to topics? Read the docs here!