Activity 2-8: Inversion

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There are some transformations of the plane we know all about:

Reflection

Rotation

Enlargement

A less well-known one is inversion.

The idea here is that you map

the point Ato the point B

where OAOB = k2

(often k = 1.)

Notice that if A maps to B, then B maps to A, so if you invert twice, you get back to where you started.

Any transformation that obeys this rule we call an involution.

Task: which of reflection, rotation and enlargement are involutions?

Using coordinate geometry, putting O as the origin, we have:

If you know about polar coordinates, then inversion is even more simply defined,

as the transformation taking the point (r, θ)to the point (k2/r, θ)

What happens to circles and straight lines in the diagram when they are inverted?

By the definition of inversion, it is clear thata straight line through the origin inverts to itself.

Task: which points on the line through the origininvert to themselves?

A straight line not through the origin?

So a line not through the origin mapsto a circle through the origin.

By the involution property of inversion, we can say the reverse too:

a circle through the origin maps to a straight line not through the origin.

Which leaves...what happens to a circle not through the origin?

It maps to another circle,

also not through the origin.

So to summarise:

A straight line through O maps to itself.

A circle not through O maps to a circle not through O.

A straight line not through O maps to a circle through O.

A circle through O maps to a straight line not through O.

If you consider a straight line as being a circle with infinite radius, then we could say that property of being a circle

is invariant under inversion.

Fine, you could say, but what use is inversion?

Inversion gives a new way of proving things.

Given a red circle inside a blue circle,

show that if you can form a chainof black circles

that meet up exactly, then it matters not where

you start your chain, and the chain will always

contain the same number of circles.

Helpful fact: given one circle inside another, by choosing our centre and radius of inversion carefully,

we can invert the circles into two concentric circles.

Task: experiment with this

Autograph file.The green circle

is the circleof inversion,

the dotted red circle inverts to the red circle,

while the dotted blue circle invertsto the blue circle.

What happens to the circles in the chain after inversion?

Things that touchmust invert to

things that touch.Circles not through Oinvert to circles not

through O. So the black circles

must invert to the chain on the left.

It is now completely obvious that if a chain is formed, it can start anywhere,

and the number of circles in the chain will always be the same.

Now remember that inverting again gets you back to your starting diagram,

And our proof is complete.

This result is known as the Steiner Chain.

Jakob Steiner (1796 – 1863)

The Shoemaker’s Knife

To prove: that the centre of circle Cn

is ndn away from the line k,where dn is the diameter of the circle Cn.

Invert centre O with radius chosen

so that Cn is invariant

under inversion (C2 here).

A and B invert to straight lines perpendicular to k. They must touch C2, C’1 and C’0.

The only possible diagram is as above, from which it is clear

that the centre of C2 is 2d2 above the line k.

C1 and C0 must invert to circles not through O touching A’ and B’.

We can argue similarly

for larger n.

With thanks to:

Kenji Kozai and Shlomo Libeskindfor their article

Circle Inversions and Applications to Euclidean Geometry.

Carom is written by Jonny Griffiths, mail@jonny-griffiths.net