Conformal Mapping Wedge to Plate

[member 428835]

Hi PF!

Does anyone know the conformal map that takes a wedge of some interior angle $\alpha$ into a half plane? I'm not talking about the potential flow, just the mapping for the shape.

Thanks!

 

[marcusl]

Yes, it’s Schwarz-Christoffel

 

[andrewkirk]

Say the wedge $W$ is bounded by the curves, in polar coordinates, $c1:\theta=0, c2:\theta=\alpha$ and $c3:r=R$ with the last one, excluding its end points, included in $W$ but the other two not.

Then we could try to map $c3$ to the horizontal axis of the half plane, via a function $g$ that maps $r$ to $y$ and another function $f$ that maps $\theta$ to $x$.

We require $f:(0,\alpha)\to(-\infty,\infty)$ and $g:(0,R]\to [0,\infty)$.

The logit function is a natural choice for $f$. Any cdf for a random variable distribution with support on the whole real line should also work.

For $g$ we can take any function $h$ that is zero at $r=R$ and approaches $\infty$ as $r\to 0$ with no turning points or inflections between, (eg $h(r)=\tan ((1-r/R)\pi/2))$. I expect we want $\frac{dg}{dr}(0)=0$, to which end we could deduct $((1-r/R)\pi/2)\frac{dh}{dr}(0)$ from $h$. So a function that might work is $y = g(r) = \tan((1-r/R)\pi/2) - (1-r/R)\pi/2$.

 

[member 428835]

Thank you both!