216 Chapter^5We note that if lal = 1, so that a = eic>, the transformation reduces to
a pure rotation (of amplitude a). In particular, if a = i, every figure is
rotated 11· /2 radians. For instance, the straight line z = b + ct goes into
the perpendicular line w = ib + ict.
On the other hand, if the coefficient a is real and positive, then a = 0,
and the transformation reduces to a pure homotecy. For the special case
a = 1, we have w = z and every point maps into itself. This is called
the identity transformation. The identity transformation also results from
(5.1) when b = 0.
5.3 THE LINEAR FUNCTION w = az + b (a ::/-0)
The mapping defined by this function may be thought as a composition
of (5.2) and (5.1). In fact, if we let z1 = az, we. have w = z1 + b. Thus,
in geometric terms, the linear transformation is composed of a homotecy
of ratio lal, a rotation of amplitude Arg a and a translation determined
by the vector b.
Therefore, under the linear transformation w = az + b, straight lines go
into straight lines, circles into circles, and angles are preserved in magnitude
and orientation. In general, each geometric figure is mapped into a directly
similar figure.
If we consider any three distinct points z 1 , z 2 , z 3 in the plane and the
three corresponding points w1 = az1 + b, w2 = az2 + b, wa = aza + b,
we obtain
wa - w1 = a(za - z1) and W3 - W2 = a( Z3 - Z2)so that
(5.3-1)which means: The simple ratio of three points z1, z2, Z3 is equal to the
simple ratio of the corresponding points w 1 , w 2 , w 3 • In other words, the
simple ratio of three points is invariant under a linear transformation.
By using the notation (z1, z 2 , z 3 ) to denote the ratio Z3 - zif z 3 - z 2 ,
equation (5.3-1) can be written as
(5.3-2)By considering the triangles with vertices z1, z 2 , z 3 and w 1 , w 2 , w 3 , re-
spectively, we see that (5.3-1) expresses the proportionality of two pairs of
corresponding sides as well as the equality of the angles formed by those
sides, and hence the similarity of the triangles, since (5.3-1) implies that
lwa - w1I lz3 - z1I
----=
lwa - wzl lza - z2I