Inverse Points with Respect to a Circle

z and z* are inverse points w.r.t. the circle |z−z₀| = r if they lie on the same ray from z₀ and the product of their distances from z₀ equals r². Equivalently: (z−z₀)(z̄*−z̄₀) = r². The inverse of a point inside the circle lies outside, and vice versa.

Inverse points are the complex plane version of the reflection of a point in a circle — the analogue of how a mirror reflects a point in a line.

Geometric Definition

Two points $z$ and $z^*$ are inverse with respect to the circle $|w - z_0| = r$ if:

1. They lie on the same ray from the centre $z_0$: $\arg(z - z_0) = \arg(z^* - z_0)$
2. The product of their distances from $z_0$ equals $r^2$: $|z - z_0| \cdot |z^* - z_0| = r^2$

Algebraic Condition

From condition 1: $z^* - z_0 = \lambda(z - z_0)$ for some real $\lambda > 0$.

From condition 2: $|z - z_0| \cdot |\lambda(z - z_0)| = r^2 \implies \lambda = \dfrac{r^2}{|z - z_0|^2}$.

Substituting:

$$z^* - z_0 = \frac{r^2}{|z - z_0|^2}(z - z_0) = \frac{r^2}{(z-z_0)\overline{(z-z_0)}}(z - z_0) = \frac{r^2}{\overline{z - z_0}}$$

Taking the conjugate of both sides:

$$\bar{z}^* - \bar{z}_0 = \frac{r^2}{z - z_0}$$

Multiplying both sides by $(z - z_0)$:

$$\boxed{(z - z_0)(\bar{z}^* - \bar{z}_0) = r^2}$$

Special Case: Unit Circle

For $z_0 = 0$, $r = 1$:

$$z \cdot \bar{z}^* = 1 \implies z^* = \frac{1}{\bar{z}}$$

More generally for $|z - z_0| = r$:

$$z^* = z_0 + \frac{r^2}{\bar{z} - \bar{z}_0}$$

Key Properties

Inside maps outside: If $|z - z_0| < r$ (z inside the circle), then $|z^* - z_0| = r^2/|z - z_0| > r$ (z* outside). Inversion swaps interior and exterior.

Points on the circle are self-inverse: If $|z - z_0| = r$, then $|z^* - z_0| = r^2/r = r$, so $z^* = z$.

The centre has no inverse: As $z \to z_0$, $|z^* - z_0| \to \infty$. The inverse of the centre is the "point at infinity."

Connection to the Inversion Map

The formula $z^* = z_0 + r^2/(\bar{z} - \bar{z}_0)$ is the general inversion map (cn76 specialised to $z_0 = 0$, $r$ general). The condition $(z-z_0)(\bar{z}^*-\bar{z}_0) = r^2$ is its defining equation — one complex equation carrying both the collinearity and distance conditions simultaneously.