Determine if two line segments intersect

Write a function to tell if two line segments intersect or not

This is a very interesting geometry problem. To solve this problem, first let's consider several cases:

Intersect, not parallel

Intersect, parallel

Intersect, q lies on rs

Not intersect, not parallel

Not intersect, parallel

So, the problem becomes :
1. determine if any end point on one lines lies on the other line, if there is one, then of course two lines intersect
2. determine if two lines are parallel

For the second problem, we need to introduce a new notion: orientation. Orientation of an ordered triplet of point on a plane can be:

• clockwise;
• counterclockwise;
• parallel

Figure source: http://www.geeksforgeeks.org/check-if-two-given-line-segments-intersect/

Two lines intersect if one of these conditions are satisfied:

1. (p, q, r) and (p, q, s) have different orientations AND (r, s, p) and (r, s, q) have different orientations;

Different orientation, intersect

Same orientation, not intersect

Determine the orientation, we need to use a little bit our high school or college math: cross product.

 Ok, now the problem becomes very clear. 

/**
  * check if two line segments pq and rs intersect with each other
  * @param p
  * @param q
  * @param r
  * @param s
  * @return
  */
 public static boolean hasIntersection(Point p, Point q, Point r, Point s) {
  if (p == null || q == null || r == null || s == null)
   throw new NullPointerException("Null points!");
  if (p.equals(q) || r.equals(s))
   throw new IllegalArgumentException("Not a line!");
  int ori1 = orientation(p, q, r);
  int ori2 = orientation(p, q, s);
  int ori3 = orientation(r, s, p);
  int ori4 = orientation(r, s, q);
  if (ori1 != ori2 && ori3 != ori4)
   return true;
  if (ori1 == 0 && isOnSegment(p, q, r))
   return true;
  if (ori2 == 0 && isOnSegment(p, q, s))
   return true;
  if (ori3 == 0 && isOnSegment(r, s, p))
   return true;
  if (ori4 == 0 && isOnSegment(r, s, q))
   return true;
  return false;
 }
 /**
  * check if point r is on line segment pq
  * @param p
  * @param q
  * @param r
  */
 private static boolean isOnSegment(Point p, Point q, Point r) {
  return r.x <= Math.max(p.x, q.x) && r.x >= Math.min(p.x, q.x) 
    && r.y <= Math.max(p.y, q.y) && r.y >= Math.min(p.y, q.y);
 }
 /**
  * return the orientation of the triplet
  * 0: parallel
  * 1: clockwise
  * 2: counterclockwise
  * cross product
  * http://en.wikipedia.org/wiki/Cross_product
  * pq & pr
  * @param p
  * @param q
  * @param r
  * @return
  */
 private static int orientation(Point p, Point q, Point r) {
  double orientation = (p.x - q.x) * (p.y - r.y) - (p.y - q.y) * (p.x - r.x);
  if (orientation == 0)
   return 0;
  return orientation > 0 ? 1 : 2;
 }