§2.1 The Extended Complex Plane and its Spherical Representation

2.1 The Extended Complex Plane and its Spherical Representation

All complex numbers form a field that extends the real number field. A complex number $𝛼 + i 𝛽$ can be visualized on a rectangular plane as the point $(𝛼, 𝛽)$ , with two axes: the real axis and the imaginary axis. It is well known that any complex number also has the polar form $𝑟 e^{i 𝜃} = 𝑟 (cos 𝜃 + i sin 𝜃)$ .

The point at infinity, $\infty$ , extends $ℂ$ to

\hat{ℂ} = ℂ \cup {\infty} .

The following arithmetic operations are defined: for all $𝑎 \in ℂ$ ,

𝑎 + \infty = \infty + 𝑎 = \infty,

and for all $𝑏 \in ℂ ∖ {0}$ ,

𝑏 \cdot \infty = \infty \cdot 𝑏 = \infty, \frac{𝑎}{\infty} = 0 .

Let

𝑆^{2} = {(𝑥_{1}, 𝑥_{2}, 𝑥_{3}) \in ℝ^{3} : 𝑥_{1}^{2} + 𝑥_{2}^{2} + 𝑥_{3}^{2} = 1} .

There exists a stereographic projection of $𝑆^{2}$ onto $\hat{ℂ}$ . For every point other than $(0, 0, 1)$ , there is a corresponding complex number

𝑧 = \frac{𝑥_{1} + i 𝑥_{2}}{1 - 𝑥_{3}} . (2.1 . 1)

This correspondence between $ℂ$ and $𝑆^{2} ∖ {(0, 0, 1)}$ is injective. In fact, the inverse can be solved for:

{| 𝑧 |}^{2} = \frac{1 - 𝑥_{3}^{2}}{{(1 - 𝑥_{3})}^{2}} = \frac{1 + 𝑥_{3}}{1 - 𝑥_{3}},

which results in

𝑥_{3} = \frac{{| 𝑧 |}^{2} - 1}{{| 𝑧 |}^{2} + 1},

and consequently,

𝑥_{1} = ℜ𝔢 (𝑧) (1 - 𝑥_{3}) = \frac{𝑧 + \overline{𝑧}}{{| 𝑧 |}^{2} + 1},

𝑥_{2} = ℑ𝔪 (𝑧) (1 - 𝑥_{3}) = \frac{𝑧 - \overline{𝑧}}{i {| 𝑧 |}^{2} + i} .

By letting $\infty$ correspond to $(0, 0, 1)$ , the bijection is complete. The sphere $𝑆^{2}$ is also called the Riemann sphere. The region given by the disk $| 𝑧 | < 1$ corresponds to $𝑥_{3} < 0$ , and the region $| 𝑧 | > 1$ corresponds to $𝑥_{3} > 0$ .

We will now give a geometric visualization of this projection. Let $𝑧 = 𝑥 + i 𝑦$ . Then we obtain

𝑥 = \frac{𝑥_{1}}{1 - 𝑥_{3}} and 𝑦 = \frac{𝑥_{2}}{1 - 𝑥_{3}} .

Therefore,

𝑥 : 𝑦 : 1 = 𝑥_{1} : 𝑥_{2} : 1 - 𝑥_{3} .

It follows that the points $(0, 0, 0)$ , $(𝑥, 𝑦, 1)$ , and $(𝑥_{1}, 𝑥_{2}, 1 - 𝑥_{3})$ are collinear in $ℝ^{3}$ . Under the linear map

𝐯 \mapsto 𝐯 (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{matrix}) + (0, 0, 1),

we get that $(0, 0, 1)$ , $(𝑥, 𝑦, 0)$ , and $(𝑥_{1}, 𝑥_{2}, 𝑥_{3})$ are collinear. In other words, this correspondence is a central projection with center $(0, 0, 1)$ , projecting the points from $𝑆^{2} ∖ (0, 0, 1)$ onto $ℂ$ . Let this center correspond to $\infty$ . In this representation, $\infty \in \hat{ℂ}$ is no longer considered to be “special”.

It is worth noting that in several geometric contexts, an alternative paradigm exists where we let $𝑆$ be the sphere centered at $(0, 0, \frac{1}{2})$ of diameter $1$ , and project points from the north pole $(0, 0, 1)$ onto the horizontal plane of tangency. The corresponding equations are then

𝑥_{1} = \frac{ℜ𝔢 (𝑧)}{{| 𝑧 |}^{2} + 1}, 𝑥_{2} = \frac{ℑ𝔪 (𝑧)}{{| 𝑧 |}^{2} + 1}, 𝑥_{3} = \frac{{| 𝑧 |}^{2}}{{| 𝑧 |}^{2} + 1} .

The forward projection remains unchanged. Lastly, we define the upper half-plane; for the following sections, let

ℍ^{+} = {𝑧 \in ℂ : ℑ𝔪 (𝑧) > 0}