session we consider the transformation
where a,b,c,d are complex constants satisfying
transformation is clearly a combination of translation,
rotation, stretching or
contraction, and inversion.
transformation is called bilinear because it is
left-hand side is seen to be linear in z and w.
expect circles to map into circles; that is,
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
Notice that this correspondence is continuous in the sense
The inverse transformation
shows a one-to-one correspondence from the extended
complex plane onto the extended complex plane.
equation of any line or circle in the complex plane can be
written in the form
and C are real numbers with AC<|B|
the equation reduces to
Letting z=x+iy and B=
we get (Then
represents a line.
equation of a circle with center
the equation above clearly represents a circle.
extended complex plane, lines can regarded as circles
matter of fact, in the case
replacing z by
the resulting equation into
is a circle. Hence we group circles and lines in one
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
implies that the result is true in this case.
then by dividing the numerator and denominator in the
general bilinear transformation by
we may write the transformation as
Making the substitution
which can be regarded as an inversion
followed by a magnification and rotation
followed by a translation