# Determination of a Joukowski airfoil chord (demonstration)

I'm currently studying Aerodynamics, and one thing that I noticed is that the maximum and minimum $$x$$ coordinate of the airfoils (which are necessary to compute the chord) on the transformed plane (let it be $$z(x,y)$$) correspond to the transformed intersection points of the circumference on the original plane (let it be $$\zeta(\xi,\eta)$$) with the real axis. I don't find any proof of this statement, and so I tried to do it by myself. The problem is that it got too complicated to be solved analytically (too many different powers of trignometric functions). So I'm requesting someone to try demonstrate this too. I think that there's a simpler way to do it.

There's my introduction to the problem:

Consider the original circumference defined on the complex $$\zeta$$ plane:

where $$a$$ is the circumference radius, and $$b$$ the intersection of the circumference with the real positive axis, $$\xi$$. The parameter $$\beta$$ is the angle between the horizontal line and the line that links $$\zeta_0$$ to $$b$$. The center of the circumference is:

$$\zeta_0=-b\varepsilon+ia\sin(\beta)=-b\varepsilon+i(1+\varepsilon)b\tan(\beta)$$

So, this circumference is defined by:

$$\zeta=-b\varepsilon+b\frac{1+\varepsilon}{\cos(\beta)}\left(e^{i\theta}+i\sin(\beta)\right),\hspace{15pt}\theta \in [0,2\pi]$$

Now I need to show that for the Joukowski transform $$z=\zeta+\frac{b^2}{\zeta}$$ the $$x$$ coordinate (real coordinate on the plane $$z$$) has a maximum on $$\zeta=b$$ (intersection of the circunference with the positive real axis) and a minimum on $$\zeta=-b(1+2\varepsilon)$$, (intersection of the circunference with the negative real axis).