Elementary Functions 269
and to the ray arg z = B+rr corresponds the same hyperbola. The semiaxes
of the hyperbola are
a= I cos Bl, b = I sin Bl
so the semifocal distance is c = J cos^2 B + sin^2 B = 1. Thus all hyperbolas
corresponding to different values of B are confocal with foci at the points
(-1, 0) and ( +l, 0). However, for B = 0 the hyperbola degenerates into
the ray u 2:: 1, v = 0, described twice as r increases from 0 to +oo, and
for B = 7r the hyperbola degenerates into the ray u :::; -1, v = 0, also
describes twice as r increases from 0 to +oo. For B = ±rr /2 the hyperbola
degenerates into the imaginary axis.
The circles with center at the origin and the rays issued from the origin
form two orthogonal families of curves, as well as their images (confocal
ellipses and hyperbolas) at corresponding points, exception being made of
those lines intercepting at the points (-1, 0) and (1, 0). These are "critical"
points where the mapping ceases to be isogonal. In fact, the angle between
the circle lzl = 1 and either ray B = 0 or B = 7r is duplicated by the
transformation.
Next we consider a circle 'Yin the z-plane passing through z = -1 and
such that the point z = l lies within 'Y· We wish to construct the image of 'Y
under the J oukowski transformation. For this purpose we superimpose the
z-and w-planes and begin by finding the image of 'Y under the reciprocal
transformation w = 1/ z. This is the circle 1' in Fig. 5.22. If C is the
center of 'Y and A = (-1, 0), the center C' of 1' lies in the intersection of
CA with the symmetric of the line CO with respect to the imaginary axis.
Of course, the circle 1' contains the point A since the reciprocal of -1 is
-1. Now consider a number of points on"(, say the six points P1, P2, ... ,
P 6 , as well as the corresponding points P{, P~, ... , P~ on "(^1 • Then it is
an easy matter to determine by vector addition and scalar multiplication
the points w =^1 / 2 (z + l/z) = 1/iOPk +OP/.) on the curve r, the image
of 'Y under the Joukowski transformation. This curve is called Joukowski's
aerofoil or profile. t
A modification of (5.16-5) in the form
:~~~ = (:~~rleads to a class of curves called Karman-Trefjtz profiles [10].
tThe connection between Joukowski's and the wing profiles that occur in actual
practice amounts to a simple geometrical resemblance.