function dist_btwn_pts(pt1, pt2) {
  return Math.pow(Math.pow(pt1.x - pt2.x,2) + Math.pow(pt1.y - pt2.y,2) , 0.5);
}

function angle_btwn_lines(line1, line2){
  var vector1 = {x: line1.x2 - line1.x1, y: line1.y2 - line1.y1};
  var vector2 = {x: line2.x2 - line2.x1, y: line2.y2 - line2.y1};
  var angle = (Math.atan2(vector2.y, vector2.x) - Math.atan2(vector1.y, vector1.x));
  if (angle < 0) { angle = angle + 2 * Math.PI; }
  return angle;
}

function btwn(a, b1, b2) {
  var tol = 1e-6;
  if ((a >= b1 - tol) && (a <= b2 + tol)) { return true; }
  if ((a >= b2 - tol) && (a <= b1 + tol)) { return true; }
  return false;
}

function line_line_intersect(line1, line2) {
  var x1 = line1.x1, x2 = line1.x2, x3 = line2.x1, x4 = line2.x2;
  var y1 = line1.y1, y2 = line1.y2, y3 = line2.y1, y4 = line2.y2;
  var pt_denom = (x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4);
  var pt_x_num = (x1*y2 - y1*x2) * (x3 - x4) - (x1 - x2) * (x3*y4 - y3*x4);
  var pt_y_num = (x1*y2 - y1*x2) * (y3 - y4) - (y1 - y2) * (x3*y4 - y3*x4);
  if (pt_denom == 0) { return "parallel"; }
  else { 
    var pt = {'x': pt_x_num / pt_denom, 'y': pt_y_num / pt_denom}; 
    if (btwn(pt.x, x1, x2) && btwn(pt.y, y1, y2) && btwn(pt.x, x3, x4) && btwn(pt.y, y3, y4)) { return pt; }
    else { return "not in range"; }
  }
}

function path_line_intersections(pathEl, line2, n_segments) {
  var pts = [];
  var int_line_segs = [];
  for (var i=0; i<n_segments; i++) {
    var pos1 = pathEl.getPointAtLength(pathLength * i / n_segments);
    var pos2 = pathEl.getPointAtLength(pathLength * (i+1) / n_segments);
    var line1 = {x1: pos1.x, x2: pos2.x, y1: pos1.y, y2: pos2.y};
    var pt = line_line_intersect(line1, line2);
    if (typeof(pt) != "string") {
      pts.push(pt);
      int_line_segs.push(line1);
    }
  }
  return d3.zip(pts,int_line_segs);
}

function first_path_line_intersection(pathEl, line, n_segments) {
  var start_pt = {x: line.x1, y: line.y1};
  var int_pts_and_segs = path_line_intersections(pathEl, line, n_segments);
  if (int_pts_and_segs.length > 0) {
    var distances = int_pts_and_segs.map(function(pt) { return dist_btwn_pts(start_pt, pt[0]); });
    var min_pt = d3.zip(int_pts_and_segs,distances).sort(function(a,b){ return a[1] - b[1]; })[0][0];
    return min_pt;
  } else {
    return 'none';
  }
}

function compute_reflection(pt, angle, int_seg ) {
  var pt2 = {x: pt.x + int_seg.x2 - int_seg.x1,
         y: pt.y + int_seg.y2 - int_seg.y1}
  var rot_pt_x = pt.x + (pt2.x - pt.x) * Math.cos(angle) - (pt2.y - pt.y) * Math.sin(angle);
  var rot_pt_y = pt.y + (pt2.x - pt.x) * Math.sin(angle) + (pt2.y - pt.y) * Math.cos(angle);
  var l = dist_btwn_pts(pt, pt2);
  return { dx: (rot_pt_x - pt.x ) / l, dy: (rot_pt_y - pt.y) / l };
}