The map that trades circles for lines

Fix a circle of radius $R$ about a point $O$. Send every other point $P$ to the point $P'$ on the ray from $O$ through $P$ whose distance satisfies $|OP| \cdot |OP'| = R^2$. That is the whole definition, and almost nothing about it looks promising: it is not a rigid motion, it does not preserve distance, it does not preserve area, and it is undefined at one point of the plane.

What it does preserve is the family of circles and lines, taken together as one family, and the angle between any two curves that cross. Those two facts are enough to turn a class of geometry problems from hard into easy, and the transformation between the two is one picture.

What the definition already says #

Three consequences fall straight out of $|OP| \cdot |OP'| = R^2$, and none of them needs any geometry beyond the equation.

A point on the mirror circle stays exactly where it is, since $|OP| = R$ forces $|OP'| = R$. A point inside goes outside and a point outside comes in, since one factor of a product fixed at $R^2$ falls exactly when the other rises. And applying the map twice returns every point to itself, because $|OP''| = R^2 / |OP'| = |OP|$ and the ray has not changed. The map is its own inverse, which is why the word inversion does not mean the opposite of something.

The centre $O$ has no image at all: as $P$ approaches it, $|OP'|$ grows without bound. Nothing can be assigned to $O$ that keeps the map continuous, and the usual repair is to add a single point at infinity and declare that $O$ and it are exchanged. That is a real construction rather than a dodge — the sphere with one point removed is the plane, so the plane with one point added is a sphere, and on the sphere inversion is a perfectly ordinary map with nothing missing.

The construction in the right-hand panel above is worth having because it is a reason rather than a formula. From an interior point $P$, draw the chord perpendicular to $OP$; it meets the circle at $T$; the tangent at $T$ meets the ray at $P'$. The triangles $OPT$ and $OTP'$ share the angle at $O$ and both have a right angle, so they are similar, and

$$\frac{|OP|}{|OT|} = \frac{|OT|}{|OP'|},$$

which is $|OP| \cdot |OP'| = R^2$ rearranged. The generator computes the constructed point and the algebraic one separately and checks that they coincide, which is the site’s habit applied to a construction rather than to a number.

Circles and lines are one family #

The claim that does the work is this: the image of a circle is a circle, unless the circle passes through the centre of inversion, in which case it is a line.

The arithmetic is quick. Put $O$ at the origin. A circle with centre $c$ and radius $r$ is the set of points with $|z|^2 - 2 z \cdot c + |c|^2 - r^2 = 0$. Applying $w = R^2 z / |z|^2$ and substituting gives, after clearing denominators,

$$(|c|^2 - r^2)\,|w|^2 - 2R^2\, w \cdot c + R^4 = 0 .$$

If $|c|^2 \ne r^2$ — that is, if the circle misses the origin — this is again the equation of a circle, with centre $R^2 c / (|c|^2 - r^2)$ and radius $R^2 r / \bigl| |c|^2 - r^2 \bigr|$. If $|c| = r$, the quadratic term vanishes and what is left is linear: a straight line. Run the same substitution on a line and a circle through the origin comes back.

So a line is not a degenerate case to be apologised for; it is a circle through the point at infinity, and inversion moves the point at infinity to $O$. On the sphere the statement is that inversion carries circles to circles, with no exception at all, and the plane’s exception is an artefact of having deleted a point.

The trap in the same picture, and it catches everybody once. The centre of the image circle is not the image of the centre. The two marked dots in the first panel above are those two points, and the caption prints the distance between them. Inversion carries shapes to shapes and carries no distinguished point of a shape to the corresponding point of its image — a fact that follows from the map not being an affine one, and one that a drawing makes obvious in a way a formula does not.

The angle survives #

The second property is the one that makes inversion a tool. Two curves crossing at some angle have images crossing at the same angle.

A map preserving angles is called conformal, and conformality is exactly what makes a transformation usable in a problem whose statement is about tangency, perpendicularity or angle: those hypotheses survive the move, so a configuration can be carried to an easier position, solved there, and carried back.

Why does the angle survive? The cleanest argument is a limit. Near a point $P$ at distance $d$ from $O$, inversion acts almost like a reflection in the mirror circle composed with a uniform scaling by $R^2/d^2$ — the further part of a small figure is shrunk slightly more than the nearer part, but as the figure shrinks the discrepancy vanishes faster than the figure does. In the limit the local behaviour is a similarity, and similarities preserve angles. The one twist is that the map reverses orientation along each ray, so it reverses the sense of an angle while keeping its size: it is anticonformal in the same way that a reflection is.

One consequence deserves its own sentence, because it is what most applications lean on. A circle that crosses the mirror circle at right angles is carried to itself — the caption above measures this, giving the same radius before and after about the same centre. Such a circle is fixed as a set, though its points move along it, exactly as a line of reflection is fixed as a set while its two sides are exchanged. That is the beginning of a whole geometry: the circles orthogonal to a fixed circle behave like the straight lines of a plane where the parallel postulate fails, and inversion in them plays the role of reflection.

A hard problem made easy #

Here is the pay-off, and it is the standard advertisement for the method because nothing else does the job so quickly.

Given two circles, one inside the other, a Steiner chain is a ring of circles each tangent to both boundaries and to its two neighbours, with the ring closing up. Whether such a chain exists looks like a hard question about the two boundary circles, and drawing one by hand is a nuisance.

Now invert. Choose the centre of inversion at one of the two limiting points of the pair — the two points that are inverse with respect to both circles at once, which exist whenever the circles are nested and disjoint. Inverting there sends both boundary circles to concentric circles, and in the concentric case the question is trivial: $n$ equal circles fit round an annulus with inner radius $a$ and outer radius $b$ exactly when

$$\sin\frac{\pi}{n} = \frac{b - a}{b + a},$$

which says the chain circle’s radius is the sine of half the angle it subtends. Inverting back turns the concentric ring into the lopsided one, tangency by tangency, because inversion preserves tangency — two curves touching at a point cross at angle zero, and zero is preserved.

Steiner’s porism is the corollary that makes the theorem famous: if a chain closes for one starting position, it closes for every starting position. In the concentric picture that is obvious, since rotating the annulus changes nothing; and obvious after an inversion is a legitimate way for something to be true. The dashed ring in the figure is a second chain started half a step round, and its tangencies are measured as well.

The generator builds the ring concentric and then inverts it, which would be worthless as evidence if the tangencies were also checked in the concentric frame, where they are arithmetic. They are checked on the drawn circles instead: centre distance against sum of radii for neighbours, against difference for the containing circle. The picture is what is verified.

The distance formula, and Ptolemy for free #

Inversion does not preserve distance, but it distorts it in a way with a formula, and the formula is short:

$$|P'Q'| = \frac{R^2\,|PQ|}{|OP|\cdot|OQ|}.$$

The proof is the similar triangles again: $OPQ$ and $OQ'P'$ share the angle at $O$, and $|OP| \cdot |OP'| = |OQ| \cdot |OQ'|$ makes the sides about that angle proportional, so the triangles are similar and the ratio of the remaining sides follows.

That formula converts a statement about four points into a statement about three, and the conversion proves Ptolemy’s theorem in a line. Take four points $A, B, C, D$ in order on a circle and invert about $A$. The circle through all four passes through the centre of inversion, so it becomes a line, and $B', C', D'$ land on it in order. Three collinear points in order satisfy an addition:

$$|B'C'| + |C'D'| = |B'D'| .$$

Now substitute the distance formula into each term and multiply through by $|AB|\,|AC|\,|AD|/R^2$. The result is

$$|AB| \cdot |CD| + |BC| \cdot |AD| = |AC| \cdot |BD| ,$$

which is Ptolemy: in a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides.

Push $D$ off the circle and the images stop being collinear. Three points that are not in a line satisfy the triangle inequality strictly, and the same substitution turns that into $|AC| \cdot |BD| < |AB| \cdot |CD| + |BC| \cdot |AD|$. So the equality case of Ptolemy’s inequality is exactly the concyclic case — and the whole of that statement is the triangle inequality, viewed from the right point.

This is what people mean when they call a transformation a change of coordinates for a problem. Nothing new was proved about circles; a hard-looking identity was moved to a place where it is the most familiar inequality in mathematics, and moved back.

Where it fails, and what it costs #

Three limitations, each of which explains a place where the method does not apply.

It destroys one point, and sometimes that point matters. Inverting about a point on a figure sends that part of the figure to infinity. Usually this is the intention — sending a circle to a line is exactly that — but it means the centre must be chosen with the conclusion in mind, and a problem whose statement is symmetric in several points may have no inversion that respects the symmetry.

It does not preserve straightness, area, or the centre of anything. A triangle does not invert to a triangle; a midpoint does not invert to a midpoint; the ratio in which a point divides a segment is not preserved. Any hypothesis of that kind has to be converted into an angle or tangency statement before the move, or it is lost.

The distance formula is not a metric statement. Because $|P'Q'|$ depends on where $P$ and $Q$ sit relative to $O$, an inequality between two distances is not carried across unchanged, and a problem about maximising a length is generally not simplified. Ptolemy survives because the products are arranged so that the awkward factors cancel — which is a fact about that particular combination and not a general licence.

There is also a cost that is easy to miss: a proof by inversion needs an argument that the configuration after inversion is really the configuration claimed, including which side of a line things land on. The pictures here are checked by pushing sample points through the map and comparing them against the image shape’s equation, and a written proof needs the same care about incidence and betweenness.

Where it came from, and where it goes #

Inversion is usually credited to Jakob Steiner in the 1820s, with earlier appearances in work by Ptolemy in a different guise and by Apollonius in the form of the two limiting points. Its systematic development belongs to the nineteenth century, when the search for transformations preserving classes of figures became a subject in its own right — geometry as the study of the invariants of a group of maps.

Two directions lead onward. The first is the sphere, where inversion becomes an ordinary rotation seen from the wrong side, and where stereographic projection makes the circle-to-line phenomenon obvious rather than surprising: circles on the sphere through the projection point project to lines, and circles missing it project to circles. The second is the complex plane, where inversion is the map $z \mapsto R^2/\bar z$ and joins the family of Möbius transformations $z \mapsto (az+b)/(cz+d)$. Those are the maps that multiply and rotate and then divide, they carry circles and lines to circles and lines, and inversion is the reflection that generates them together with the rigid motions.

From there it becomes the standard model of hyperbolic geometry: circles orthogonal to a fixed circle as the lines, inversions in them as the reflections, and a plane in which the parallel postulate is false obtained without leaving the Euclidean page.

What the pictures cannot show #

Three things, and the first is structural. Every figure here is drawn at one radius of inversion, and the radius is irrelevant: changing $R$ rescales the whole image by a factor and changes no incidence, no tangency and no angle. A reader can only be told that, since a picture at one radius looks like a picture at another with a different zoom.

The second is the point at infinity, which is the mechanism that makes the theorem clean and cannot be drawn. Every figure showing a circle through the centre inverting to a line is drawing a finite piece of an infinite object, and the fact that the line’s two ends are the same point of the image is invisible. That identification is what makes lines and circles a single family, and the only honest picture of it is on a sphere.

The third is the failure mode the method is most often used to hide. A construction that works after inversion has to be pulled back, and the pull-back can leave a configuration in a different arrangement — the point that was inside the circle is now outside, the arc that was minor is now major. None of the figures here shows a case where that reversal changes the conclusion, because none exists in these examples; but such cases exist, and the picture will not warn anybody about them.

The ladder from here #

Sideways: stereographic projection, which is the same phenomenon with the sphere made explicit, and the inscribed angle, whose invariance is the classical fact the tangency arguments above lean on. Below: multiplication as turning, which is the arithmetic behind writing inversion as $z \mapsto R^2/\bar z$. Above: the Möbius transformations, the hyperbolic plane, and the circle packings whose rigidity theorems are stated in exactly the language this map provides — where the boundary between two regions is a curve rather than a line is one place where inversion turns a hard partition into an easy one.

What is worth carrying away #

The lasting lesson is about how a transformation earns its place. Inversion preserves almost nothing a beginner would call important — not length, not area, not straightness, not the centre of a circle — and it is nevertheless one of the sharpest tools in plane geometry, because it preserves the two things that most of the interesting hypotheses are made of: the class of circles-and-lines, and angle.

The method for finding such a tool generalises. List the hypotheses a family of problems actually uses; find the transformations that leave those hypotheses alone; then look for the position in which the conclusion becomes obvious. Steiner’s porism is the model case, and its proof is one sentence once the right position has been found: make the two circles concentric.