conceptually, you can see that the "angle part" remains the same, all we are doing is "stretching" or "shrinking" the radius.

this is why the tangent function is used. what we are doing is mapping [0,1) to [0,∞). note that between 0 and π/2 tangent takes on all positive values. so all we need to do to map [0,1) to [0,∞) is to find some function that maps [0,1) to [0,π/2). the function f(x) = (π/2)x, will do nicely, so for a function that maps [0,1) to [0,∞) we simply take:

tan○f