Question

I try to determine the angle from a point (n,m) to (0,0). Without arctan2 being available, I’m running into the problem that m can be 0, which leads to a possible division by zero.

What would be an elegant, correct solution to tackling this issue?

Answer 1

don’t use the conventional quadrants, use the branch points defined by the lines y = +/- x, and use the 1st two steps of a CORDIC-like algorithm (e.g. rotate coordinates by a known angle and keep track of how much you’ve rotated):

function atan2_substitute(x,y) {    double angle = 0;    if (x < y)    { angle = M_PI; x = -x; y = -y; }    // this guarantees that the angle is between -135 and +45 degrees     if (x < -y)    {      angle -= M_PI/2; tmp = x; x = -y; y = tmp;    }    // this guarantees that the angle is between -45 and +45     angle += atan(y/x);     if (angle > M_PI)       angle -= 2*M_PI;    // fails at 0,0; otherwise is accurate over the entire plane } 

the reason for doing it this way is that atan() may be more likely to be accurate for ratios y/x between -1 and +1, than for ratios greater than 1. (though a good atan() algorithm would recognize this and take the reciprocal)