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!