Bilinear transformation



In this session we consider the transformation where a,b,c,d are complex constants satisfying  ad-bc0.

This transformation  is clearly a combination of translation, rotation, stretching or

contraction, and inversion.

This transformation is called bilinear because it is equivalent to


where the left-hand side is seen to be linear in z and w.

If we expect circles to map into circles; that is,

we notice the “problem” in the variable term |and we now set our objective  to show that this special conformal transformation maps “lines and circles” into “lines and circles” in the extended complex plane as we shall see shortly.  If c=0, then implies .  If then implies that and implies that .  Notice that this correspondence is continuous in the sense that as then and as , then    The inverse transformation shows a one-to-one correspondence from the extended complex plane onto the extended complex plane.

Theorem.  The equation of any line or circle in the complex plane can be written in the form


where A and C are real numbers with AC<|B|

Proof.  For A=0  the equation  reduces to +C=0.  Letting  z=x+iy and B= we get (Then  which represents a line.

Now the equation of a circle with center and radius is or

.  Thus where

For the equation above clearly represents a circle.   

In the extended complex plane, lines can regarded as circles through

As a matter of fact, in the case replacing z by transforms the resulting equation into

which for is a circle.  Hence we group circles and lines in one category.

Since z=w-b it follows that under these three transformations “lines and circles” are mapped into “lines and circles”.  Now we show that this is also true for any bilinear transformation:

If then implies that the result is true in this case.

If then by dividing the numerator and denominator in the general bilinear transformation by , we may write the transformation as

  Making the substitution =z+d yields which can be regarded as an inversion followed by a magnification and rotation

(), followed by a translation .