inversion in a circle
Reflection across a straight line flips each point to the other side, keeping its distance to the line. Inversion in a circle is the curved cousin of that idea: it turns a circle 'inside out', swapping the inside with the outside so that points near the centre fly far away and points far away crowd in near the centre, all along the straight ray from the centre. It is the one genuinely new transformation that conformal geometry adds to the familiar slides, spins, and scalings.
Here is the precise rule, and it is easy to compute. Fix a circle with centre O and radius r. To invert a point P (other than O), travel along the ray from O through P and find the point P' on that same ray whose distance from O satisfies |OP| times |OP'| = r^2. So if P is at distance 3 from O and r = 6, then P' is at distance 36 over 3, that is 12, on the same side: close points map to far points and vice versa. Points exactly on the circle (distance r) stay put, since r times r equals r^2. The centre O has no image in the ordinary plane — it would need to go infinitely far — which is precisely why inversion is studied on the inversive plane, where O maps to the point at infinity. Inversion has three signature properties: it is its own undo (invert twice and you are back where you started), it is conformal (it preserves the angle between any two curves, though it reverses their orientation), and it maps the family of lines-and-circles to itself — a circle through O becomes a straight line, a circle not through O becomes another circle, and a line becomes a circle through O.
Inversion is the engine behind Mobius geometry: every Mobius transformation is a composition of inversions, rotations, translations, and scalings, so inversion is to conformal geometry what reflection is to Euclidean geometry. It is also a sharp problem-solving tool — many hard problems about tangent circles become easy after a well-chosen inversion straightens the circles into lines. Two cautions. First, inversion is not the same as scaling: the factor by which it stretches lengths depends on how far the point is from the centre, so it distorts shape even while preserving angles. Second, do not expect it to preserve centres: the image of a circle is a circle, but the image of the original centre is generally not the centre of the image circle.
Invert in a circle of radius r = 6 centred at O. A point P at distance 4 from O maps to P' at distance 36 over 4, that is 9, on the same ray; a point at distance 9 maps back to distance 4; a point exactly on the circle, at distance 6, stays fixed. A small circle passing through O straightens out into a straight line, which is the trick that makes inversion so useful.
|OP| times |OP'| = r^2 along the ray from O: near and far swap, the circle itself stays fixed.
Inversion preserves angles but not shape, and it does not preserve centres or radii — the image of a circle is a circle, yet the image of that circle's centre is generally not the centre of the new circle.