1. Prof. Harriet Fell Fall 2011 Lecture 9 – September 26, 2011
CS 4300 Computer Graphics
Prof. Harriet Fell
Fall 2011
Lecture 9 – September 26, 2011
January 1, 2019

2. Today’s Topics Fill: Flood Boundary Fill vs. Polygon Fill
See Photoshop for Flood Fill
2D Polygon Fill
January 1, 2019

3. Simple 2D Polygon an ordered sequence of line segments such that
each edge gi = (si, ei) is the segment from a start vertex si to an end vertex ei
ei-1 = si for 0 < i ≤ n-1 and en-1 = s0
non-adjacent edges do not intersect
the only intersection of adjacent edges is at their shared vertex
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4. Scan Line Polygon Fill
January 1, 2019

5. Parity Check Draw a horizontal half-line from P to the right.
Count the number of times the half-line crosses an edge.
1 in
4 out
7 in
7 in
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6. Polygon Data Structure
edges
xmin
ymax
1/m 
(9, 6) 1 6
8/4 
xmin = x value at lowest y
ymax = highest y
Why 1/m?
(1, 2)
If y = mx + b, x = (y-b)/m.
x at y+1 = (y+1-b)/m = (y-b)/m + 1/m.
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7. Polygon Data Structure13 12 11
10  e6 9 8
7  e4  e5
6  e3  e7  e8 5 4 3 2
1  e2  e1  e11
0  e10  e9
Edge Table (ET) has a list of edges for each scan line. 13 e6 10 e5 e3 e7 e4 e8 5 e9 e2
e11
e10 e1

8. Preprocessing the edges
For a closed polygon, there should be an even number of crossings at each scan line.
We fill between each successive pair.
count twice, once for each edge
delete horizontal edges
chop lowest pixel to only count once
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9. Polygon Data Structure after preprocessing13 12 11
10  e6 9 8
7  e4  e5
6  e3  e7  e8 5 4 3 2
1  e2  e1  e11
0  e10  e9
11  e6 10
Edge Table (ET) has a list of edges for each scan line. 13
7  e3  e4  e5
6  e7  e8 e6 10 e5 e3 e7 e4 e8 5 e9 e2
e11
e11
e10 e1

10. The Algorithm Start with smallest nonempty y value in ET.
Initialize SLB (Scan Line Bucket) to nil.
While current y ≤ top y value:
Merge y bucket from ET into SLB; sort on xmin.
Fill pixels between rounded pairs of x values in SLB.
Remove edges from SLB whose ytop = current y.
Increment xmin by 1/m for edges in SLB.
Increment y by 1.
January 1, 2019

11. Running the Algorithm ET e6 e5 e3 e7 e4 e8 e9 e2 e11 e10 11  e613 12
11  e6 10 9 8
7  e3  e4  e5
6  e7 ve8 5 4 3 2
1  e2  e11
0  e10  e9
xmin ymax 1/m
e /5
e3 1/ /3
e /5 e
e6 6 2/ /3
e /2 e
e /8
e /4
e /3
Running the Algorithm 13 e6 10 e5 e3 e7 e4 e8 5 e9 e2
e11
e10 5 10 15

12. Running the Algorithm y=0 SCB e10 e6 11 10 1/4 4 -3/4  e5 e3 e7 e913 e6 11
10 1/4 4
-3/4  10 e5 e3 e7 e9 e4 e8
11 3/8 11 8
3/8  5 e9 e2
e11
e10 5 10 15
January 1, 2019

13. Running the Algorithm y=1 SLB e2 e6 2 1 3/5 6 -2/5  e5 e3 e7 e11 e413 e6 2
1 3/5 6
-2/5  10 e5 e3 e7
e11 e4 e8 6
6 2/3 4
2/3  5 e9
e10 e2
e11
9 1/2
10 1/4 4
-3/4 
e10 e9 5 10 15
11 6/8
11 3/8 8
3/8 
January 1, 2019

14. Running the Algorithm y=2 SLB e2 e6 1 3/5 1 1/5 6 -2/5  e5 e3 e7 e1113 e6
1 3/5
1 1/5 6
-2/5  10 e5 e3 e7
e11 e4 e8
6 2/3
7 1/3 4
2/3  5 e9
e10 e2
e11
9 1/2
8 3/4 4
-3/4 
e10 e9 5 10 15
11 6/8
12 1/8 8
3/8 
January 1, 2019

15. Running the Algorithm y=3 SLB e2 e6 4/5 1 1/5 6 -2/5  e5 e3 e7 e1113 e6
4/5
1 1/5 6
-2/5  10 e5 e3 e7
e11 e4 e8
7 1/3 8 4
2/3  5 e9
e10 e2
e11 8
8 3/4 4
-3/4 
e10 e9 5 10 15
12 1/8
12 4/8 8
3/8 
January 1, 2019

16. Running the Algorithm y=4 SLB e2 e6 4/5 6 -2/5  e5 e3 e7 e11 e4 e8 813 e6
4/5 6
-2/5  10 e5 e3 e7
e11 e4 e8 8 4
2/3  5 e9
e10 e2
e11 8 4
-3/4 
e10 e9 5 10 15
12 4/8 8
3/8 
Remove these edges.
January 1, 2019

17. Running the Algorithm y=4 SLB e2 e6 2/5 4/5 6 -2/5  e5 e3 e7 e9 e413 e6
2/5
4/5 6
-2/5  10 e5 e3 e7 e9 e4 e8
12 7/8
12 4/8 8
3/8  5 e9
e11 and e10 are removed. e2
e11
e10 5 10 15
January 1, 2019

18. Running the Algorithm y=5 SLB e2 e6 2/5 6 -2/5  e5 e3 e7 e9 e4 e813 e6
2/5 6
-2/5  10 e5 e3 e7 e9 e4 e8
13 2/8
12 7/8 8
3/8  5 e9 e2
e11
e10 5 10 15
January 1, 2019

19. Running the Algorithm Remove this edge. y=6 SLB e2 e6 6 -2/5  e5 e313 e6 6
-2/5  10 e5 e3 e7 e7 e4 e8 10
9 1/2 10
-1/2  5 e9 e8 e2
e11 10 12 8 2 
e10 e9 5 10 15
13 2/8
13 5/8 8
3/8 
January 1, 2019

20. Running the Algorithm Add these edges. e3 1/3 12 1/3  e4 4 12 -2/5 13 e6 e5 10 e5 4 13  e3 e7 e4 e8
y=7
SLB e7 5
9 1/2 10
-1/2  e9 e2
e11 e8
e10 12 8 2  5 10 15 e9
13 5/8 8
3/8 

21. Non-Simple Polygons
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22. Even-odd Rule
construct a ray r in an arbitrary direction from the test point p
count number of intersections of r with the polygon. p is defined to be inside the poly if the intersection count is odd.
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23. Reasoning
each time an edge is crossed, either switch from inside to outside or outside to inside
but since poly is closed, know that ray r must end up outside
this is the method we just applied
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24. Nonzero Rule construct a ray r as before
compute all intersections of r with the poly edges gi, but this time keep track of whether the edge crossed from left to right (i.e. si on left side of r and ei on right side of r) or right to left
count +1 for left-to-right and –1 for right-to-left
p is defined to be inside the poly if and only if (iff) final count is nonzero
1/1/2019

25. Reasoning consider sweeping a point q along the perimeter of the poly
take the integral of the orientation angle of the line segment from p to q
turns out that, for one complete sweep of the polygon, the integrated winding angle will be an integer multiple of cover it)
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26. Intuition
intuition: reasonable definition of inside-ness is to check if the poly “circled around” p in CCW direction different number times than in CW direction
nonzero intersection test turns out to be a shortcut to compute (proof is a little subtle and we will not cover it)
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27. The Results Can be Different
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