Peaucellier-Lipkin Mechanism

I came across an interesting mechanism today.
The Peaucellier Lipkin Mechanism :

This mechanism gives a straight line output for a circular input.
made using the principle of inversion of circles.
                   

Here points O and A lie on a circle (input circle).
OB = OC , AB=AC=CE=EB
its obvious from its symmetry that points O, A and E are collinear.

This is how it works :
suppose there is a circle with radius r.
now take an arbitrary point (P) anywhere

There will be a point  'T' (Output) on OP which divides the line OP such that OP*OT=r^2








now  move P along an arbitrary  circle  :

Using the relation OP*OT = r^2
point T was plotted :

The yellow circle is the path traced by T

 
Now as the distance between the P's Circle and O decreases the size of circle traced by T increases
until it touches O : then at that point when P passes through O in order to keep the relationship OP*OT=r^2  true ; T becomes infinite ie; it traces a circle of infinite diameter  which to us is visible as a straight line.

                     

The MATLAB code used  can be found here.