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Can anyone advise me on how to derive a conformal map for this mapping? I am familiar with how to apply Schwarz-Christoffel from the upper half plane to the quadrilateral, but how do I then map from the quadrilateral to the sector?

Parcly Taxel
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niran90
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  • Are there any extra conditions? Not all quadrilaterals with the same four angles are similar to each other. – Maxim Nov 30 '19 at 23:29
  • @Maxim Hmm, I hadn't thought of that because I didn't think it would have any consequence to the derivation of the map. For my purposes, arbitrary lengths can be assigned to the segments as long as the angles at the vertices are satisfied. – niran90 Dec 02 '19 at 07:26

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If we start with the sector with the angle $\pi/4$, then $z^8$ maps it to $\mathbb D \setminus [0, 1)$. The Joukowsky transform maps $\mathbb D \setminus [0, 1)$ to $\mathbb C \setminus [-1, \infty)$. A branch of $\sqrt {z + 1}$ maps $\mathbb C \setminus [-1, \infty)$ to the upper half-plane. By the Schwarz-Christoffel formula, a branch of $z^{1/4} \hspace {1.5px} {_2 F_1}(1/8, 1/2; 9/8; z^2)$ maps the upper half-plane to a quadrilateral with the given angles. Then we have to take the inverse of the resulting composition of mappings.

If we want to get a quadrilateral similar to the given one, the problem is more difficult, because only three of four points on the real axis can be chosen arbitrarily.

Maxim
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  • Thanks so much for your response! I have to be honest in that I do not have a rigorous-enough mathematical background (mech. engineer in computational fluid dynamics) to be able to immediately see how I can use what you've said in order to implement this mapping in my code. But I shall do a bit more reading, based on your response.

    How would I go about deriving a 'continuous' (but not necessarily conformal) mapping for the above? Would that be a lot simpler? The map should not significantly distort/stretch shapes (so the ideal case would be conformality, but this is not necessary).

     – niran90 Dec 03 '19 at 07:36
  • You'll need to invert the mapping $w = f(z)$ numerically. You can use Newton's method ($f'$ is elementary) or solve the ODE $z' = 1/f'(z)$. – Maxim Dec 03 '19 at 16:35