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).
Although the math behind the conformal transformation may seem obscure, the mathematical process is rather direct. Necessary elements in the transformation are a transform circle ζ of radius a and nondescript vector Z extending from the origin to various positions on the perimeter of ζ. The length of vector Z from the origin to the circumference of ζ changes through the continuum of increasing positive angles θ being swept out by Z. For each angular position taken by Z, vector $b^2$/Z is poised in the interior of ζ at corresponding angle -θ. The airfoil contour in z(x,y) space is defined by the vector sum Z + $b^2$/Z. The airfoil chord is defined by the transformed maximum- and minimum-abscissa crossings of the transform circle. The Z + $b^2$/Z vector sum at these points defines the end points of the abscissa-length spanned by the airfoil in z(x,y) space. In many instances, the airfoil chord can be completely within the profile of the airfoil contour. More highly cambered airfoils may have a portion of the chord exterior to the concave lower portion of the contour. Camber is simply the curved-profile form of the airfoil.
The first included illustration summarizes all salient and relevant features needed to answer the question. The circle center $ζ_0$ has an abscissa given as -bε. The circle having this center and positive-abscissa crossing at b is of radius a, with radius-center $ζ_0$ making shown angle β with the abscissa through b. For this discussion, however, β is taken as the positive angle –β + $\pi$. Consequently, the distance from the maximum crossing at b, to the abscissa of the center at ξ=-bε, is bε+b. The minimum crossing in relation to the origin may be readily determined as ξ=-b-$2$bε or, exactly, -b($1-2$ε). In relative terms, a•cos(β) = bε+b, the distance from the maximum at b to the center abscissa at -bε, and that $2$a•cos(β)=$2$(bε+b), the distance from b to the circle minimum-abscissa crossing. Consequently, this minimum is also obtained as ξ=-a•cos(β)-bε.
When Z is aligned coincident with the abscissa at $\theta$=$0$ or $\pi$, $b^2$/Z is also coincident with the abscissa. Consequently, Z is obtained directly by previous determination that ξ=b or ξ=-b-$2$bε at these locations, and summation of Z with $b^2$/Z for each case gives maximum and minimum abscissa values of the airfoil.
Exactly, what are these values? We can see that when Z is aligned with the positive abscissa, Z has length b, as does $b^2$/Z. The airfoil coordinate is the sum Z + $b^2$/Z, namely $2$b. Consequently, z(x,y) becomes ($2$,$0$) if b is assigned a value of $1$. If b is greater or lesser, all of the dimensions and functions in the transformation space are changed proportionally. Consequently, changes in b give no change to the appearance of the airfoil. Commonly, b is assigned a fixed value of $1$ because associations with b are then greatly simplified.
When $\theta$=$\pi$, Z is directed at the circle minimum-abscissa. At this position Z + $b^2$/Z transforms into the airfoil minimum-abscissa. Assigning b as $1$ gives the circle minimum as Z = -$1$–$2$ε. The vector sum becomes (-$1$–$2$ε) + ($1$/(-$1$–$2$ε)). We can also look at this in trigonometric terms. The connection of β to $1$+ε gives the same result in a different form yielding first Z = -a•cos(β)-ε, and then counterpart $1$/Z = (a•cos(β)-ε)$^2$/(-a•cos(β)-ε). However, *(a•cos(β)-ε)$^2$ is really $1$. Consequently, the vector sum for the minimum becomes (-a•cos(β)-ε) + (($1$/(-a•cos(β)-ε)). As one can see, these are two apparently different forms giving the same minimum x-axis value for the airfoil. Really, they are the same.
Not all of the results are as easily determined as those at b, the circle abscissa-maximum crossing. This is because the circle maximum-abscissa crossing is at a fixed point, b, in relation to other aspects of the circle connected to b that can be varied. This simplification is not the case with the circle minimum-abscissa crossing. The ξ coordinate -ε for abscissa of the circle center determines the coordinate of the circle minimum-abscissa crossing. The negative value of ε is determined by the radius a and angle β anchored at b. To keep -ε constant, an increase in β requires an increase in a. The distance -ε of circle-center $ζ_0$ from the η - axis will vary most notably with changes in radius a. Consequently, β varied within constrained limits will change the relationship of a fixed-value radius a and the resulting negative value of ε. The most important aspect to keep in mind is that in the Joukowski transformation, a fixed value of the center ordinate, -ε, will always give the same result for the airfoil minimum-abscissa crossing.
A total picture of the Joukowski transformation can be gained from the following sources:
K. Pohlhausen, in Theory of Functions as Applied to Engineering Problems: Technology Press, M.I.T, 1933. B. Two-Dimensional Fields of Flow, pp. 103 - 125. In this reference, the unit circle is used as the basis for airfoil camber and radius-ratio definition of the transform circle. Polhausen, however, is not specific about definition of the unit circle or of the transform circle in this way.
Milne-Thomson, L.M., Theoretical Hydrodynamics, 5th ed.: The Macmillan Company, 1968. 7.30. Further investigation of the Joukowski transformation, p. 195 ff. Special attention should be paid to 7.31 regarding geometrical relationships in the transformation.
The following illustrations provide a visual demonstration of this answer for -ε held to a constant value of -ε = $-0.111$. The circle minimum-abscissa crossing is at ξ = -$1$+$2$ε having a value of $-1.223$ as shown by the small circle marking that point on the transform-circle perimeter. The airfoil minimum crossing for the transformed point in z(x,y) space is at z(x) = $-2.041$. Angle β and radius a have the following values in the first illustration (more cambered airfoil), $2.889$ and $1.148$, and in the second illustration (less cambered airfoil), $3.016$ and $1.12$. For the calculation of these examples, β is measured counter-clockwise to the center of radius a from the positive-directed ξ-axis at b. In both illustrations, b has a value of $1$, the corresponding airfoil maximum x-coordinate is at $2$. The airfoil chord has a length of $4.041$.
salient and relevant features in the transformation $\beta$ and radius a are $2.889$ and $1.148$, respectively $\beta$ and radius a are $3.016$ and $1.120$, respectively