algorithm - Find the arc of the intersection between circle and rectangle -

i need find largest arc created intersection of circle , rectangle. have center of circle, radius , coordinates of rectangle, , need find angle of intersection points center of circle.

i have code works, computes solution iterating points of circumference, , wondering if there's more elegant way calculate solution using trigonometry instead of "brute force".

that's code:

import 'dart:math';  class circlethesquare {   final point     _circlecenter;   final int       _circleradius;   final rectangle _box;     circlethesquare(this._circlecenter, this._circleradius, this._box);     map<string, double> arc {     map res = new map();      double angle = .0;     double anglein;     double angleout;     double increment = 1.0;      while (true) {       if (angle > 360.0 && anglein == null) {         break;       }        // finds point of intersection (next points inside        // of rectangle).       if (!_isoutside(angle) && _isoutside(angle - increment)) {         anglein = angle;       }        // finds next intersection (next points outside        // of rectangle).       if (anglein != null &&           _isoutside(angle + increment) && !_isoutside(angle)) {          angleout = angle;          // adds arc result there's not previous largest arc.         if (res["in"] == null ||             angleout - anglein > res["arc"]) {            res["in"]  = anglein;           res["arc"] = angleout - anglein;         }         anglein = null;         angleout = null;       }       angle += increment;     }      // if there's no intersections.      // -- simplicity, assume     //    rectangle , circle intersect or circle      //    inside of rectangle).     if (res["in"] == null) {       res = {"in" : 0.0, "arc" : 360.0};     }      return res;   }     bool _isoutside(double a) {     var res;      double cx = _circlecenter.x + (_circleradius * cos(a * (pi / 180)));     double cy = _circlecenter.y + (_circleradius * sin(a * (pi / 180)));      bool hout = cx < _box.left || cx > _box.left + _box.width;     bool vout = cy < _box.top || cy > _box.top + _box.height;      if (hout || vout) {       res = true;     } else {       res = false;     }      return res;   } }   main() {   circlethesquare = new circlethesquare(new point(250, 250), 100,                                           new rectangle(0,0,500,500));    print(a.arc); // {in: 0.0, arc: 360.0}    circlethesquare b = new circlethesquare(new point(450, 250), 100,                                           new rectangle(0,0,500,500));    print(b.arc); // {in: 60.0, arc: 240.0}    circlethesquare c = new circlethesquare(new point(420, 420), 100,                                           new rectangle(0,0,500,500));    print(c.arc); // 4 intersections, returns largest arc:                           // {in: 127.0, arc: 196.0} } 

1. shift coordinates make circle zero-centered (box.left = box.left - circlecenter.x etc) simplicity
2. find intersections of circle sides of rectangle. example, left side solve (box.left)^2 + y^2 = radius^2, check point lies on side, add intersection point list
3. sort intersection points angle (probably, automatically provided side check order), find largest angle interval arcs inside rectangle