Circle Inversion: The Reflection That Swaps Inside and Outside

The rule: one product, and nothing else

Fix a circle ω with centre O and radius r > 0. Inversion in ω sends a point P ≠ O to the unique point P′ on the ray from O through P for which |OP| × |OP′| = r². In vectors that is P′ = O + r²(P − O)/|P − O|², and if we put O at the origin of the complex plane it is z → r²/z̄, the conjugate reciprocal scaled by r².

Three consequences fall straight out of the product rule. First, inversion is an involution: |OP′| = r²/|OP|, so applying it again returns |OP| and the original point. Second, it swaps the inside and the outside: |OP| < r forces |OP′| > r and vice versa. Third, it fixes ω point by point: |OP| = r gives |OP′| = r²/r = r, and since P′ is on the same ray, P′ = P.

Take r = 1 with O at the origin. The point (4, 0) goes to (0.25, 0); the point (0.25, 0) goes back to (4, 0); the point (0.6, 0.8), which sits on the unit circle, does not move at all. The one point with no image is O itself, because no finite P′ satisfies 0 × |OP′| = r². The standard repair is to work in the inversive plane, the ordinary plane plus a single point at infinity, and to declare that inversion swaps O and ∞. Topologically the inversive plane is a sphere, which is exactly the Riemann sphere of complex analysis.

Why the product is r squared: the tangent construction

The product rule is not an arbitrary definition dropped from the sky; it is what a ruler-and-compass construction produces. Let P lie outside ω. Draw a tangent line from P touching ω at T, and drop the perpendicular from T onto the segment OP, meeting it at Q. Then Q is the inverse of P.

Here is the reason, and it is a complete proof rather than an illustration. The radius OT meets the tangent at a right angle, so triangle OTP has a right angle at T. Triangle OQT has a right angle at Q by construction. The two triangles share the angle at O, so they are similar, which gives |OQ| / |OT| = |OT| / |OP|. Since |OT| = r, this is |OQ| × |OP| = r² — precisely the inversion rule. Running the construction backwards inverts a point inside the circle: erect the chord through P perpendicular to OP, take either endpoint T, and the tangent to ω at T meets the ray OP at P′.

Numerically, with r = 1 and |OP| = 1.45, the tangent length is √(1.45² − 1) = √1.1025 = 1.05, and the foot of the perpendicular lands at |OP′| = 1/1.45 = 0.690. The same statement is the power of the point O in disguise: if a line through O meets a circle at A and B, then |OA| × |OB| is the same number for every such line, and provided O lies outside that circle, inverting with r² equal to that number maps the circle to itself. The hypothesis is doing work: if O lies inside, the common product is ρ² − d², and inverting with that radius carries the circle to its reflection in O rather than to itself.

Lines and circles stop being different things

The reason inversion is worth learning is that it refuses to distinguish a circle from a line. In inversive geometry the two are bundled into one object, the generalized circle or cline: a circle in the plane, or a line together with the point at infinity. Inversion permutes clines, and there are exactly four cases, laid out in the table above.

The interesting case has a short proof. Let L be a line not through O, let F be the foot of the perpendicular from O to L, with |OF| = h, and let F′ be the inverse of F, so |OF′| = r²/h. Take any other point X on L. Because |OF| × |OF′| = |OX| × |OX′| = r², we have |OF| / |OX| = |OX′| / |OF′|, and the triangles OFX and OX′F′ share the angle at O. They are therefore similar, so the angle OX′F′ equals the angle OFX, which is 90°. A point that sees the segment OF′ at a right angle lies on the circle with diameter OF′. So the image of L is the circle through O of diameter r²/h, and running the argument backwards gives the converse.

Concretely, with r = 1 and O at the origin, the vertical line x = 2 inverts to the circle of centre (0.25, 0) and radius 0.25. Check a second point: (2, 2) has |OX|² = 8, so it maps to (0.25, 0.25), and that point is a distance 0.25 from (0.25, 0), as required. This is what the animation shows when a square grid dissolves into a bouquet of circles: each grid line that misses the centre becomes a circle through the centre, and lines at distance 0.6, 1.2 and 1.8 produce image circles of radius 0.833, 0.417 and 0.278. The two grid lines that do pass through O stay dead straight. The moving picture is an accurate rendering of the map, but the proof is the similar-triangle argument above; the picture is evidence, not a substitute for it.

What survives inversion, and what does not

Angles survive. Inversion is anticonformal: it preserves the size of the angle between two curves and reverses its sense. The quickest argument is complex-analytic. The map z → 1/z is holomorphic away from 0 with derivative −1/z², which never vanishes, so it is conformal; inversion in the unit circle is that map followed by conjugation, and conjugation is a reflection, which flips orientation while leaving angle magnitudes alone. A 60° counterclockwise crossing therefore comes out as a 60° clockwise crossing.

Tangency survives, because tangency is the statement that two clines meet at angle zero. Two circles that touch invert to two clines that touch. There is one genuine exception and it is worth stating precisely: if the two circles are tangent at the centre O itself, their images are two parallel lines, which in the inversive plane touch at the point at infinity rather than anywhere you can draw.

Distance does not survive, but it distorts by a formula clean enough to be a tool in its own right: for any P, Q ≠ O,

|P′Q′| = r² |PQ| / (|OP| × |OQ|)

which follows from the similarity of triangles OPQ and OQ′P′. Section five turns this single identity into a proof of Ptolemy's theorem.

Centres do not survive. This trips up nearly everyone the first time. Inversion maps a circle to a circle, but it does not map the old centre to the new centre. Take r = 1, O at the origin, and the circle of centre (2, 0) and radius 1, which meets the x-axis at 1 and at 3. Those two points invert to 1 and 1/3, so the image is the circle of centre (2/3, 0) and radius 1/3. But the image of the centre (2, 0) is (1/2, 0), and 1/2 ≠ 2/3. In general a circle of centre C and radius ρ with d = |OC| ≠ ρ maps to the circle of centre O + k(C − O) and radius |k|ρ, where k = r²/(d² − ρ²) and the denominator is the power of O with respect to the circle.

That formula also settles which circles are fixed. If a circle crosses ω at right angles then d² = r² + ρ², so k = r²/r² = 1 and the image is the circle itself. Apart from ω itself, which is fixed point by point, the circles orthogonal to ω are exactly the circles that inversion leaves alone as sets, and that fact is the engine of the Poincaré disc model of hyperbolic geometry.

Turning hard circle problems into easy line problems

Ptolemy's theorem and inequality. For any four points A, B, C, D in the plane, |AC| × |BD| ≤ |AB| × |CD| + |AD| × |BC|, with equality exactly when the four are concyclic (or collinear) in that cyclic order. Invert at A with any radius k. The distance formula turns each of the three products into a single image distance, and after dividing through by |AB| × |AC| × |AD| / k² the inequality becomes |B′D′| ≤ |B′C′| + |C′D′| — the triangle inequality, with equality precisely when C′ lies on the segment B′D′. A four-point metric statement has been reduced to a fact about three points on a line. Sanity check on a 3 by 4 rectangle: the diagonals are 5, so the left side is 25 and the right side is 3 × 3 + 4 × 4 = 25.

Steiner's porism (Jakob Steiner, 1826). Put one circle inside another, not concentric, and run a chain of circles each tangent to both and to its two neighbours. Steiner's result is that if the chain ever closes up after n circles, it closes up from every starting position. Any two non-intersecting circles can be inverted into a pair of concentric circles, and between two concentric circles the chain is obviously rigid under rotation. Inverting back carries that rotational freedom to the original configuration.

The Pappus chain and the arbelos. Pappus of Alexandria, around 320 AD, proved with great effort that the centre of the n-th circle in the chain inscribed in an arbelos sits a height 2nρ_n above the base line, where ρ_n is that circle's radius. Invert in a circle centred at the cusp and the chain becomes a stack of equal circles between two parallel lines, where counting is trivial.

Apollonius' problem — construct a circle tangent to three given circles, which has eight solutions in general position — and Feuerbach's theorem of 1822, that the nine-point circle touches the incircle and all three excircles, both have standard proofs that begin by choosing an inversion centre that collapses the hard part into lines.

Magnus, Kelvin, and a linkage that draws a straight line

The individual facts are old — Apollonius of Perga was studying loci defined by ratios of distances around 200 BC — but inversion as a transformation to be applied wholesale is a nineteenth-century idea. Adolphe Quetelet published worked cases in 1825, and Jakob Steiner was using inversive arguments in the mid-1820s. The first general treatment normally credited is Ludwig Immanuel Magnus (1790–1861), in his 1831 paper in Crelle's Journal für die reine und angewandte Mathematik. Julius Plücker (1834), Giusto Bellavitis (1836) and the Dublin geometers John Stubbs and John Ingram (1842–43) developed it further, and August Möbius gave the analytic theory of circle transformations in 1855.

The Peaucellier–Lipkin linkage. In 1864 the French army officer Charles-Nicolas Peaucellier built a seven-bar linkage that performs inversion mechanically; Yom Tov Lipman Lipkin found it independently and published in 1871, and Sylvester popularised it in an 1874 Royal Institution lecture. Two bars of length ℓ run from a fixed pivot O to opposite corners of a rhombus of side s, and the other two corners are P and P′. The geometry forces |OP| × |OP′| = ℓ² − s², a constant: the linkage is an inversion of radius √(ℓ² − s²). Pin P to a crank that sweeps a circle through O, and P′ traces an exactly straight line. That solved a problem that had defeated steam-engine designers for a century; Watt's famous linkage of 1784 only approximates a straight line.

Kelvin's method of images. In 1845 William Thomson, later Lord Kelvin, wrote to Liouville describing how to solve electrostatics problems by inversion. A point charge q sitting a distance a from the centre of a grounded conducting sphere of radius R produces the same external field as the charge q together with an image charge −qR/a placed at distance R²/a from the centre — at the inverse point. With R = 10 cm and a = 25 cm, the image is −0.4q at 4 cm from the centre. The underlying analytic statement is the Kelvin transform: in n dimensions, if u is harmonic then so is |x|²⁻ⁿ u(r²x/|x|²) on the inverted domain, with the prefactor disappearing in the plane.

Where circle inversion lives today

Möbius transformations. Every map z → (az + b)/(cz + d) with ad − bc ≠ 0 is a composition of an even number of inversions in clines, and every conjugate-linear version is a composition of an odd number. Inversions are the reflections of inversive geometry, in the same way that line reflections generate the Euclidean isometries.

Hyperbolic geometry. In the Poincaré disc model the hyperbolic lines are the arcs of circles orthogonal to the boundary, and hyperbolic reflection in such a line is inversion in that circle — which, as shown above, fixes the disc because orthogonality forces the scale factor to be 1. The whole isometry group of the hyperbolic plane is generated by circle inversions.

Apollonian gaskets and Kleinian groups. Start with mutually tangent circles and invert repeatedly; the orbit is an Apollonian gasket, whose curvatures obey the Descartes circle theorem, (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²) — found by Descartes in 1643 and rediscovered by Frederick Soddy in 1936. Groups generated by inversions in several circles are the classical Schottky and Kleinian groups.

The sphere. Stereographic projection is the restriction of an inversion in three dimensions, which is why it maps circles to circles and preserves angles; under it, inversion in the unit circle of the plane corresponds to reflecting the Riemann sphere in its equatorial plane. In dimension three and above the picture becomes rigid: Liouville's theorem of 1850 says that every conformal map of a domain in Rⁿ, n ≥ 3, is a Möbius transformation, that is a composition of similarities and sphere inversions. The plane is the exception, and the Riemann mapping theorem of 1851 says how large that exception is.

What inversion in a circle of radius r centred at O does to each kind of object

Object going in	Position relative to the centre O	Object coming out	Key number
Straight line	passes through O	the same line, with O removed	fixed as a set, not point by point
Straight line	nearest point is a distance h from O	a circle through O	diameter r²/h
Circle	passes through O, diameter D	a straight line missing O	distance r²/D from O
Circle	misses O; centre distance d, radius ρ	a circle centred on the line OC — on the ray OC when O is outside it, on the opposite ray when O is inside it	radius r²ρ / |d² − ρ²|
Circle	crosses the inversion circle at right angles	itself	d² = r² + ρ², so the factor is exactly 1