Problem
Problem 1.79 Let $\varphi: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ be the map given by
$$
u=x^{2}+y^{2}+z^{2}-1, \quad v=a x+b y+c z, \quad a, b, c \in \mathbb{R}, a^{2}+b^{2}+c^{2}=1
$$
(i) Find the points at which $\varphi$ is a submersion.
(ii) Find $\varphi^{-1}(0)$.
(iii) Find the points where $\varphi$ is not a submersion, and its image.