Quadrics in projective space

Verify that in the space $\mathbb {R}\mathbb {P}^{3}$ there are exactly 8 classes of projectively equivalent quadrics (including zero and degenerate quadrics, as well as double subspaces). In a suitable homogeneous coordinate system, representatives of these classes are given by the equations: $1: \xi_{0}^{2}+\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2}=0 ;$ $2: \xi_{0}^{2}+\xi_{1}^{2}+\xi_{2}^{2}-\xi_{3}^{2}=0 ;$ $3: \xi_{0}^{2}+\xi_{1}^{2}-\xi_{2}^{2}-\xi_{3}^{2}=0$; $4: \xi_{0}^{2}+\xi_{1}^{2}+\xi_{2}^{2}=0 ;$ $5: \xi_{0}^{2}+\xi_{1}^{2}-\xi_{2}^{2}=0$; $6: \xi_{0}^{2}+\xi_{1}^{2}=0$; $7: \xi_{0}^{2}-\xi_{1}^{2}=0$; $8: \xi_{0}^{2}=0$.

Using Exercise 1 and Exercise 5.2.1, give affine depictions of the quadrics of types $1-3$.