Finding PathItem Intersection Points

If you are working with simple paths, you can implement the Live Paint trick in a script. After Live Paint > Expand, a point is created at the intersection. We can compare the original array of points and the new array to find the coordinates of the new point. I find this trick fun, but it may not work in some cases.

Because Live Paint > Expand shifts the path points slightly, the calculated coordinates of the intersection points are not quite accurate. If you need maximum coordinate accuracy.

function main() {
  if (selection.length !== 2 || selection.typename === 'TextRange') return;
  var path1 = selection[0];
  var path2 = selection[1];
  if (path1.typename !== 'PathItem' || path2.typename !== 'PathItem' ) return;

  path1.duplicate();
  path1.selected = false;

  path2.duplicate();
  path2.selected = false;

  app.executeMenuCommand('group');
  app.executeMenuCommand('Make Planet X');
  selection[0].translate(0, 0);
  app.executeMenuCommand('Expand Planet X');
  try {
    app.executeMenuCommand('ungroup');
    app.executeMenuCommand('ungroup');
  } catch (e) {}

  var diff = findDiffPoints(path2.pathPoints, selection[1].pathPoints);
  alert('Intersection points:\n' + diff.join('\n'));

  for (var i = selection.length - 1; i >= 0; i--) {
    selection[i].remove();
  }
}

function findDiffPoints(points1, points2) {
  var diff = [];
  for (var i = 0; i < points2.length; i++) {
    var found = false;
    for (var j = 0; j < points1.length; j++) {
      if (matchNum(points2[i].anchor[0], points1[j].anchor[0], 0.1) && 
      matchNum(points2[i].anchor[1], points1[j].anchor[1], 0.1)) {
        found = true;
        break;
      }
    }
    if (!found) diff.push([points2[i].anchor[0].toFixed(3), points2[i].anchor[1].toFixed(3)]);
  }
  return diff;
}

function matchNum(num1, num2, tolerance) {
  var diff = Math.abs(num1 - num2);
  return diff <= tolerance;
}

main();

I have not seen a ready-made script.  There's a well known algorithm out there to find the intersection of two cubic Bezier curvese (the Bezier subdivision algorithm), but it is not simple, and it will take some work to write it into a script.