Q for the math guys:

Is this the closed form pdf of the angle distribution of one quarter of the circle?

http://www.wolframalpha.com/input/?i=1%2Fcos%28min%28x%2Cpi%...

1/cos(min(x,pi/2-x))

plus the normalization term to make \int(..) == 1:

http://www.wolframalpha.com/input/?i=int+1%2Fcos(min(x%2Cpi%...

Constructed by arguing that the probability is directly proportional to the length of the hypotenuse.