to the entrance pupil. The remapping transforms the entrance pupil intensity P(x,y) into P1(x,y).]{}

3.  [Choose a focal plane mask diameter $a$.]{}

4.  [Compute the generalized prolate function $Prola(x,y)$ for the remapped aperture shape defined by $P1(x,y)>0$, using the focal plane mask diameter $a$. This is done iteratively as described in section \[ssec:APCMLCprinc\]]{}

5.  [Compute the amplitude ratio $Apo(x,y)=Prola(x,y)/P1(x,y)$. This is the post-apodizer amplitude transmission function. $Apo(x,y)$ is then scaled to ensure that its maximum value is equal to 1. The intensity-weighted average of $Apo(x,y)^2$ defines the coronagraph throughput for off-axis sources.]{}

Steps (3) to (5) are repeated for different values of the focal plane mask size $a$. The off-axis coronagraph throughput is computed for each choice of $a$, and the final focal plane mask size is chosen to maximize throughput. This optimization links the choice of the remapping function (step (1)) to a value of the focal plane mask radius a. For a circular unobstructed aperture, the solution would be $a=b$, for which the PIAA apodization would perfectly match the generalized prolate function. On arbitrarily shaped pupils, the focal plane mask radius is usually close to, but not equal to, $b$. Stronger apodization functions correspond to larger values for 