Inversion: The Map z → 1/z̄

The inversion map w = r²/z̄ sends every point z ≠ 0 to its inverse point w on the same ray from origin with |z||w| = r². Key property: circles and lines map to circles or lines under inversion. A circle through the origin maps to a line; a line not through the origin maps to a circle through the origin. This is the complex form of inverse points (cn50). Connects back to: Circles chapter (inversion, radical axis).

Inversion is the transformation that swaps the inside and outside of a circle while preserving the angles between curves. It maps circles and lines to circles and lines — a property that makes otherwise hard problems tractable.

Definition from Inverse Points

Two points $z$ and $z^*$ are inverse with respect to the circle $|z| = r$ if:

1. They lie on the same ray from the origin: $\arg(z^*) = \arg(z)$
2. Their distances multiply to $r^2$: $|z| \cdot |z^*| = r^2$

From condition 1: $z^* = \lambda z$ for some real $\lambda > 0$.

From condition 2: $|z| \cdot |\lambda z| = r^2 \implies \lambda = r^2/|z|^2$.

So:

$$z^* = \frac{r^2}{|z|^2} \cdot z = \frac{r^2}{z\bar{z}} \cdot z = \frac{r^2}{\bar{z}}$$ $$\boxed{w = \frac{r^2}{\bar{z}}}$$

Circles Map to Circles or Lines

Consider the general circle $z\bar{z} + \bar{a}z + a\bar{z} + b = 0$.

Under inversion $z \mapsto r^2/\bar{w}$ (i.e., $z = r^2/\bar{w}$, $\bar{z} = r^2/w$):

$$\frac{r^2}{\bar{w}} \cdot \frac{r^2}{w} + \bar{a} \cdot \frac{r^2}{\bar{w}} + a \cdot \frac{r^2}{w} + b = 0$$ $$\frac{r^4}{w\bar{w}} + \frac{r^2\bar{a}}{\bar{w}} + \frac{r^2 a}{w} + b = 0$$

Multiply through by $w\bar{w}/r^2$:

$$r^2 + \bar{a}w + a\bar{w} + \frac{b}{r^2}w\bar{w} = 0$$ $$\frac{b}{r^2}w\bar{w} + \bar{a}w + a\bar{w} + r^2 = 0$$

Case 1 — $b \neq 0$: Divide by $b/r^2$. The result is $w\bar{w} + \cdots = 0$, a circle. A circle not through the origin maps to another circle not through the origin.

Case 2 — $b = 0$: The original circle passes through the origin. The transformed equation becomes $\bar{a}w + a\bar{w} + r^2 = 0$ — linear in $w$ and $\bar{w}$, which is a straight line. A circle through the origin maps to a line.

The reverse is also true: a line (which is a circle through the origin at infinity) maps to a circle through the origin. $\blacksquare$

Angles Are Preserved

Inversion is a conformal map — it preserves the angle between two curves at their intersection. This is why problems involving angles between circles are often simplified by inverting a suitable circle to a line.

Connection to Inverse Points (cn50)

The formula $w = r^2/\bar{z}$ is the complex form of the inverse point condition derived in cn50: $(z - z_0)(\bar{z}^* - \bar{z}_0) = r^2$ for the circle $|z - z_0| = r$. When $z_0 = 0$, this gives $z\bar{z}^* = r^2$, consistent with $z^* = r^2/\bar{z}$.