Proposition

Find the image of the following points under the stereographic projection $\phi$:

• $(0, 0, -1)$,
• $(0, 0, 1)$,
• $(1, 0, 0)$,
• $(0, 1, 0)$,
• $(1, 1, 0)$.

Solution

• $(0, 0, -1) \mapsto (0, 0, 0)$,
• $(0, 0, 1) \mapsto \infty$,
• $(1, 0, 0) \mapsto (1, 0, 0)$,
• $(0, 1, 0) \mapsto (0, 1, 0)$,
• $(1, 1, 0) \mapsto (1, 1, 0)$.