+ Real x = coordinate_a_ [X_AXIS];
+ Real y = coordinate_a_ [Y_AXIS];
+
+ // We keep in the vicinity of multiples of 45 degrees here: this is
+ // where straightforward angles for straightforward angular
+ // relations are most expected. The factors of 2 employed in the
+ // comparison are not really perfect for that: sqrt(2)+1 would be
+ // the factor giving exact windows of 45 degrees rather than what we
+ // have here. It's just that 2 is likely to generate nicer code
+ // than 2.4 and the exact handover does not really matter.
+ //
+ // Comparisons here are chosen appropriately to let infinities end
+ // up in their "exact" branch. As opposed to the normal atan2
+ // function behavior, this makes "competing" infinities result in
+ // NAN angles.
+ if (y < 0.0)
+ {
+ if (2*x < -y)
+ if (-x > -2*y) // x < 0, y < 0, |x| > |2y|
+ return -180 + atan2d (-y, -x);
+ else if (-2*x >= -y) // x < 0, y < 0, |y| < |2x| <= |4y|
+ return -135 + atan2d (x - y, -y - x);
+ else // y < 0, |y| >= |2x|
+ return -90 + atan2d (x, -y);
+ else if (x <= -2*y) // x > 0, y < 0, |y| <= |2x| < |4y|
+ return -45 + atan2d (x + y, x - y);
+ // Drop through for y < 0, x > |2y|
+ }
+ else if (y > 0.0)
+ {
+ if (2*x < y)
+ if (-x > 2*y) // x < 0, y >= 0, |x| > |2y|
+ return 180 - atan2d (y, -x);
+ else if (-2*x >= y) // x < 0, y >= 0, |y| < |2x| <= |4y|
+ return 135 - atan2d (x + y, y - x);
+ else // y >= 0, |y| >= |2x|
+ return 90 - atan2d (x, y);
+ else if (x <= 2*y) // x >= 0, y >= 0, |y| < |2x| < |4y|
+ return 45 - atan2d (x - y, x + y);
+ // Drop through for y > 0, x > |2y|
+ }
+ else
+ // we return 0 for (0,0). NAN would be an option but is a
+ // nuisance for getting back to rectangular coordinates. Strictly
+ // speaking, this argument would be just as valid for (+inf.0,
+ // +inf.0), but then infinities are already an indication of a
+ // problem in LilyPond.
+ return (x < 0.0) ? 180 : 0;
+ return atan2d (y, x);