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I developed a method to transform a sphere in the complex region to the real region.
Similar to the Joukowsky airfoil, further development would result in analytic solutions for lift on a finite wing.
These figures demonstrate the ability of the method to model a finite wing using quaternions which map a sphere to a wing.
You can find the derivation of the equations and general write up in this document.
You can find the code I wrote which implements these equations on Github.
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