1. Computational Geometry
Hwangryol Ryu
Dongchul Kim

2. Computational Geometry
Intersection of line segment
Basic geometric objects
Point : ( x, y )
Line : {(x1,y1) , (x2,y2) }
Line segment : size of line is given
(x2,y2)
(x2,y2)
(x1,y1)
(x1,y1)

3. Basic geometric objects(contd)
Polygon
Convex Non-convex
Convex : Each indegree less than 180

4. Intersection of line segment
Input : a pair of line segments
Output : yes – if they intersect
no – otherwise
How do we know that intersection(x,y) exits between two lines?
(x, y)

5. Intersection of line segment(…contd)
Algorithm
Find equation of first segment
Y=m1x+c1
Find equation of second segment
Y=m2x+c2
Find intersection of two lines : (x,y)
If ( x1 < x < x2 ) && ( x3 < x < x4 )
Then return (Intersection exits)
Else
return (No intersection exits)
(x2,y2)
(x1,y1)
(x, y)
(x3,y3)
(x4,y4)

6. Intersection of line segment(…contd)
Is there a pair of line segments intersecting each other?
Naive algorithm: Check each two segments for intersection. Complexity: O(n2).

7. Plane Sweep – Algorithm
Event is any end point or intersection point.
Sweep the plane using a vertical line.
Maintain two data structures:
Event priority queue – sorted by x coordinate.
Sweep line status – Stores segments currently intersected by sweep line and maintain Red-Black Tree to support data structure.

8. Plane Sweep – Algorithm
Problem: Given n segments in the plane, compute all their intersections.
Assume:
No line segment is vertical.
No two segments are collinear.
No three segments intersect at a common point.
Event is any end point or intersection point.
Sweep the plane using a vertical line.
Maintain two data structures:
Event priority queue – sorted by x coordinate.
Sweep line status – Stores segments currently intersected by sweep line, sorted by y coordinate.

9. Plane Sweep - Basic Idea
We are able to identify all intersections by looking only at adjacent segments in the sweep line status during the sweep.
Theorem: Just before an intersection occurs, the two relevant segments are adjacent to each other in the sweep line status.

10. Plane Sweep - Basic Idea(contd)
In practice: Look ahead: whenever two line segments become adjacent along the sweep line, check for their intersection to the right of the sweep line.

11. Plane Sweep – Algorithm
Initialization:
Add all segments’ endpoints to the event queue (O(n log n)).
Sweep line status is empty.
Algorithm proceeds by inserting and deleting discrete events from the queue until it is empty.

12. Plane Sweep – Algorithm
Event A: Beginning of segment
Insert segment into sweep line status.
Test for intersection to the right of the sweep line with the segments immediately above and below. Insert point (if found) into event queue.
Complexity: n such events, O(log n) each  O(n log n) total.

13. Plane Sweep – Algorithm
Event B: End of segment
Delete segment from sweep line status.
Test for intersection to the right of the sweep line between the segments immediately above and below. Insert point (if found) into event queue.
Complexity: n such events, O(log n) each  O(n log n) total.

14. Plane Sweep – Algorithm
Event C: Intersection point
Report the point.
Swap the two line relevant segments in the sweep line status.
For the new upper segment – test it against its predecessor for an intersection. Insert point (if found) into event queue.
Similar for new lower segment (with successor).
Complexity: k such events, O(logn) each  O(klogn) total.

15. Plane Sweep – Example Ex1) s3 e1 s4 s2 s1 s0 Sweep Line Statusb4
Ex1) s3 e1 s4 a2 a4 b3 s2 b1 b0 s1 b2 s0 a1 a0
Sweep Line Status
Event Queue
s0,s1,s2,s3
a4, b1, b2, b0, b3, b4

16. Plane Sweep – Example s3 e1 s4 s2 s1 s0 Insert s4 to SLSb4 s3 e1 s4 a2 a4 b3 s2 b1 b0 s1 b2 s0 a1 a0
Insert s4 to SLS
Test s4-s3 and s4-s2. Add e1 to EQ
s0,s1,s2, s4, s3
b1, e1, b2, b0, b3, b4
Action
Sweep Line Status
Event Queue

17. Plane Sweep – Example s3 e1 s4 s2 e2 s1 s0 Delete s1 from SLS Actionb4 s3 e1 s4 a2 a4 b3 s2 e2 b0 s1 b1 b2 s0 a1 a0
Delete s1 from SLS
Test s0-s2. Add e2 to EQ
s0,s2,s4,s3
e1, e2, b2, b0, b3, b4
Action
Sweep Line Status
Event Queue

18. Plane Sweep – Example s3 e1 s4 s2 e2 s1 s0 Swap s3 and s4 . Actionb4 s3 e1 s4 a2 a4 b3 s2 e2 b0 s1 b1 b2 s0 a1 a0
Swap s3 and s4 .
Test s3-s2.
s0,s2,s3,s4
e2, b2, b0, b3, b4
Action
Sweep Line Status
Event Queue

19. Plane Sweep – Complexity Analysis
Total time complexity: O((n+k) logn). If kn2 this is almost like the naive algorithm.
Event queue: heap
Sweep line status: balanced binary tree
Total space complexity: O(n+k).