          Bilinear transformation Sessions  In this session we consider the transformation where a,b,c,d are complex constants satisfying  ad-bc≠0. 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 +C=0, 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  . 