Geometry and Trigonometry 633So we want to bring the original equation of the cardioid into this form. First, we change
it to
r=a·2 cos^2θ
2;
then we take the square root,
√
r=√
2 acosθ
2.
Multiplying by
√
r, we obtainr=√
2 a√
rcosθ
2,
or
r−√
2 a√
rcos
θ
2= 0.
This should look like the equation of a circle. We modify the expression as follows:
r−√
2 a√
rcosθ
2=r(
cos^2θ
2+sin^2θ
2)
−
√
2 a√
rcosθ
2+ 1 − 1
=
(
√
rcos
θ
2) 2
−
√
2 a√
rcos
θ
2+ 1 +
(
√
rsin
θ
2) 2
− 1.
If we seta=2, we have a perfect square, and the equation becomes
(√
rcosθ
2− 1
) 2
+
(√
rsinθ
2) 2
= 1 ,
which in complex coordinates reads|
√
z− 1 |=1. Of course, there is an ambiguity in
taking the square root, but we are really interested in the transformationφ, not inφ−^1.
Therefore, we can chooseφ(z)=z^2 , which maps the circle|z− 1 |=1 into the cardioid
r= 2 ( 1 +cosθ).
Remark.Of greater practical importance is the Zhukovski transformationz→^12 (z+^1 z),
which maps the unit circle onto the profile of the airplane wing (the so-called aerofoil).
Because the Zhukovski map preserves angles, it helps reduce the study of the air flow
around an airplane wing to the much simpler study of the air flow around a circle.
623.Letx+y=s. Thenx^3 +y^3 + 3 xys=s^3 ,so3xys− 3 xy=s^3 −1. It follows that
the locus is described by
(s− 1 )(s^2 +s+ 1 − 3 xy)= 0.