1. Computational Geometry for the Tablet PC
CSE 490ra

2. Overview Computational Geometry on the Tablet PC Geometric primitives
SUB: Ask for submission of a picture
Computational Geometry on the Tablet PC
Geometric primitives
Intersections
Polygons
Convexity

3. Tablet Geometry Basic structure – Stroke: sequence of points
SUB: What is Points[6.4]?
Basic structure – Stroke: sequence of points
Himetric coordinates
Sampled 150 times per second
Coordinates stored in an array Points

4. Computational Geometry
Algorithms for geometric computation
Numerical issues with coordinates
Importance of degenerate cases
Examples of degenerate cases
Three lines intersecting at a point
Segments overlapping
Three points co-linear

5. Basic geometry Point Line segment Distance Basic Test p (p1, p2)
Dist(p1, p2)
Basic Test
LeftOf(p1, p2, p3)
CCW(p1, p2, p3) p p1 p3 p2 p1

6. Distance computation Avoid computing square root when possible
Example: find closest point to q
closestSoFar = p1;
d = Dist(p1, q);
for k = 2 to n {
d1 = Dist(pk, q)
if (d1 < d){
closestSoFar = pk;
d = d1; }

7. Counter Clockwise Test
CCW(p1, p2, p3)
public static bool CcwTest(Point p1, Point p2, Point p3){
int q1 = (p1.Y - p2.Y)*(p3.X - p1.X);
int q2 = (p2.X - p1.X)*(p3.Y - p1.Y);
return q1 + q2 < 0; }
ASIDE: Give derivation based on the determinant

8. Segment intersection Find intersection of (p1,p2) and (p3,p4)
Q = ap1 + (1-a)p2
Q = bp3 + (1-b)p4
Solve for a, b
Two equations, two unknowns
Intersect if 0 < a < 1 and 0 < b < 1
Derived points
In general, try to avoid computing derived points in geometric algorithms

9. Problem
SUB
Tell students segments are not co-linear
Determine if two line segments (p1, p2) and (p3,p4) intersect just using CCW Tests

10. Making intersection test more efficient
Take care of easy cases using coordinate comparisons
Only use CCW tests if bounding boxes intersect

11. Computing intersections
SUB
Ignore degenerate cases
Given k strokes, find all intersections
Given a stroke, find self intersections
Break into segments, and find all pairwise intersections
How many self intersections can a single stroke with n points have?

12. Segment intersection algorithm
Run time O(nlog n + Klog n) for finding K intersections
Sweepline Algorithm 3 6 5 1 2 7 4

13. Sweepline Algorithm Event queue Move sweepline to next event
Start Segment (S2)
End Segment (E2)
Intersection (I2,4)
Move sweepline to next event
Maintain vertical order of segments as line sweeps across
Start Segment
Insert in list
Check above and below for intersection
End Segment
Remove from list
Check newly adjacent segments for intersection
Intersection
Reorder segments

14. Sweepline example 3 6 5 1 2 7 4

15. Activity: Identify when each of the intersections is detectedB

16. Polygons Sequence of points representing a closed path
Simple polygon – closed path with no self intersections

17. Polygon inclusion test
Is the point q inside the Polygon P?

18. Convexity
Defn: Set S is convex if whenever p1, and p2 are in S, the segment (p1, p2) is contained in S

19. Convex polygons P = {p0, p1, . . . pn-1} P is convex if
CCW(pi, pi+1, pi+2) for all I
Interpret subscripts mod n
Also holds for CW (depending on how points are ordered)

20. Problem: Test if a point is inside a convex polygon using CCW Tests
SUB: Describe an algorithm using CCW Tests

21. Convex hull
Smallest enclosing convex figure
Rubber band “algorithm”

22. Compute the Convex Hull
SUB: Students draw the hull
Compute the Convex Hull

23. Algorithms Convex hull algorithms: O(nlog n) Insertion algorithm
Illustrate insertion with diagram on right
Algorithms
Convex hull algorithms: O(nlog n)
Related to sorting
Insertion algorithm
Gift Wrapping (Jarvis’s march)
Divide and Conquer
Graham Scan

24. Convex Hull Algorithms Gift wrapping

25. Divide and Conquer

26. Graham Scan Polar sort the points around a point inside the hull
Scan points in CCW order
Discard any point that causes a CW turn
If CCW advance
If !CCW, discard current point and back up

27. Polar sort the red points around q (Start with p, CCW order)
SUB: Students draw the hull
Polar sort the red points around q (Start with p, CCW order) p q
SUB: Use the result for a first illustration of the algorithm

28. Graham Scan Algorithm Stack of vertices If CCW(x, y, z)
Possible hull vertices
z – next vertex
y – top of stack
x – next on stack
If CCW(x, y, z)
Push(x)
If (! CCW(x, y, z))
Pop stack x y z x z y

29. GS Example to walk through

30. Student submission: Give order vertices are discard in the scanp

31. Application of Convex Hull
Construct a Convex Hull Selection Lasoo
Maintain convex hull of pen stroke for selection
Operations on packet event in RTS
New packet is outside of the hull
Update the hull
Packet is inside the hull
Computation to make it more efficient when pen leaves the hull

32. New Packet outside the hull
SUB: Describe what to do
New Packet outside the hull

33. New Packet inside the hull
SUB: Give ideas what to do – emphasize that I don’t have an answer in mind
SUB: Describe what to do
New Packet inside the hull