I thought "a" was the half width of the road in his equation, but that can't be the case since for reasonable values of "r" and "a" the term sqrt(r^2-a^2) would be the square root of a negative number.
y = sqrt[ (d-r)^2 - a^2 ]
In the above equation y is the vertical distance from the cross street to the TA point, given a required path distance d, a street half width a, and turning radius r. This is for a path that starts at point A and ends at a point along the edge of the cross street even with the center of the turning radius (same horizontal coordinate as point C), and passes through the point that is the intersection of the curb edges.
A path that follows the circular arc around the street intersection and ends at point C will be slightly longer, so the value of y solving his equation will be slightly smaller than the solution to the above equation.
I think if you (or he) uses the value of y given by the above equation as a starting point in your iterative solver, convergence will happen much more quickly. But I don't understand his equation because I don't know what "a" is, or what "y" is for that matter.