www.carom-maths.co.uk Activity 2-10: Inversion
There are some transformations of the plane we know all about: Reflection Enlargement Rotation
A less well-known one is inversion. The idea here is that you map the point A to 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 that a straight line through the origin inverts to itself. Task: which points on the line through the origin invert to themselves? A straight line not through the origin?
So a line not through the origin maps to 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.
A straight line through O inverts to itself. So to summarise: A straight line through O inverts to itself. A straight line not through O inverts to a circle through O. A circle through O inverts to a straight line not through O. A circle not through O inverts to a circle 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 chain of 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 circle of inversion, the dotted red circle inverts to the red circle, while the dotted blue circle inverts to the blue circle.
What happens to the circles in the chain after inversion? Things that touch must invert to things that touch. Circles not through O invert 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.
Jakob Steiner (1796 – 1863) 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.
A and B invert to straight lines A’ and B’ perpendicular to k. Invert centre O with radius chosen so that Cn is invariant under inversion (C2 here). A and B invert to straight lines A’ and B’ perpendicular to k. They must touch C2, C’1 and C’0. C1 and C0 must invert to circles not through O touching A’ and B’. The only possible diagram is as above, from which it is clear that the centre of C2 is 2d2 above the line k. We can argue similarly for larger n.
Carom is written by Jonny Griffiths, mail@jonny-griffiths.net With thanks to: Kenji Kozai and Shlomo Libeskind for their article Circle Inversions and Applications to Euclidean Geometry. Carom is written by Jonny Griffiths, mail@jonny-griffiths.net