Inversion — Definition, Formula & Examples

Inversion is a geometric transformation that maps each point to a new point along the same ray from a fixed center, so that the product of their distances from the center equals a constant. Points close to the center get sent far away, and points far from the center get sent close in.

Given a circle of inversion with center $O$ and radius $r > 0$ , the inverse of a point $P \neq O$ is the unique point $P'$ on ray $\overrightarrow{OP}$ such that $OP \cdot OP' = r^2$ . This defines a bijection on the plane minus $\{O\}$ that maps the interior of the circle (excluding $O$ ) to its exterior and vice versa, while every point on the circle itself is fixed.

Key Formula

OP' = \frac{r^2}{OP}

Where:

$O$ = Center of the circle of inversion
$r$ = Radius of the circle of inversion
$OP$ = Distance from the center to the original point P
$OP'$ = Distance from the center to the image point P'

How It Works

To find the inverse of a point, draw a ray from the center

O

through the point

P

. Compute the distance

OP

, then place

P'

on that same ray at distance

\frac{r^2}{OP}

from

O

. If

P

is inside the circle,

P'

lies outside, and if

P

is outside,

P'

lies inside. Points on the circle map to themselves. A remarkable property: inversion sends lines and circles to lines or circles, which makes it a powerful tool for simplifying problems involving tangent circles or collinear points.

Worked Example

Problem: A circle of inversion has center

O

and radius

r = 6

. Find the image of a point

P

that lies on the ray from

O

at a distance

OP = 4

Step 1: Apply the inversion formula to find the distance from

O

to the image

P'

OP' = \frac{r^2}{OP} = \frac{6^2}{4} = \frac{36}{4} = 9

Step 2: Place

P'

on the same ray

\overrightarrow{OP}

, at distance 9 from

O

. Since

OP = 4 < 6 = r

, the original point was inside the circle, and

OP' = 9 > 6

, confirming the image is outside.

Answer: The image

P'

lies on ray

\overrightarrow{OP}

at a distance of 9 from

O

Why It Matters

Inversion is a key technique in competition mathematics and advanced geometry courses for converting difficult circle-tangency problems into simpler configurations of lines. It also appears in complex analysis, where inversion corresponds to the map

z \mapsto \frac{1}{z}

, and in physics when modeling electric fields via the method of image charges.

Common Mistakes

Mistake: Confusing inversion with reflection, assuming the image is on the opposite side of the circle.

Correction: In inversion, the image

P'

is always on the same ray from

O

through

P

, not on the opposite side. Only the distance changes according to

OP' = r^2 / OP