CS 4300
Computer Graphics
Prof. Harriet Fell
Fall 2012
Lecture 9 – September 24, 2012
April 8, 2015
College of Computer and Information Science, Northeastern University
1
Today’s Topics
• Fill: Flood Boundary Fill vs. Polygon Fill
 See Photoshop for Flood Fill
• 2D Polygon Fill
April 8, 2015
College of Computer and Information Science, Northeastern University
2
Simple 2D Polygon
• an ordered sequence of line segments
g , g ,..., g n  2
0
1
n1
• 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
4/8/2015
College of Computer and Information Science, Northeastern University
3
Scan Line Polygon Fill
April 8, 2015
College of Computer and Information Science, Northeastern University
4
Parity Check
Draw a horizontal half-line from P to the right.
Count the number of times the half-line crosses an edge.
April 8, 2015
1
in
4
7
out
in
7
in
College of Computer and Information Science, Northeastern University
5
Polygon Data Structure
edges
xmin
ymax
1/m

(9, 6)
1
6
xmin = x value at lowest y
8/4

(1, 2)
ymax = highest y
Why 1/m?
If y = mx + b, x = (y-b)/m.
x at y+1 = (y+1-b)/m = (y-b)/m + 1/m.
April 8, 2015
College of Computer and Information Science, Northeastern University
6
Polygon
Data Structure
13
12
11
10  e6
Edge Table (ET) has a list of
edges for each scan line.
9
8
7
 e4
 e5
6
 e3
 e7
13
 e8
5
e6
10
e3
4
3
e5
e4
e7
e8
5
2
1
 e2
 e1
0
 e10
 e9
e2
 e11
0
e9
e11
e1
e10
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
April 8, 2015
delete
horizontal
edges
chop lowest pixel
to only count
once
College of Computer and Information Science, Northeastern University
8
Polygon
Data Structure
13
12
after preprocessing
11  e6
10  e6
Edge Table (ET) has a list of
edges for each scan line.
9
8
7
 e3
e4 e4
e5 e5

6
 e7
e3 e8
e7  e8

5
13
e6
10
e3
4
3
e5
e4
e7
e8
5
2
1
 e2
0
 e10  e9
e2
 e1
e11 e11
0
e9
e11
e1
e10
The Algorithm
1. Start with smallest nonempty y value in ET.
2. Initialize SLB (Scan Line Bucket) to nil.
3. While current y ≤ top y value:
a.
b.
c.
d.
e.
April 8, 2015
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.
College of Computer and Information Science, Northeastern University
10
ET
Running the
Algorithm
13
12
11
 e6
10
9
8
7
6
 e3  e4  e5
 e7 ve8
5
4
3
2
1
0
 e2  e11
 e10 e9
xmin
e2 2
e3 1/3
e4 4
e5 4
e6 6 2/3
e7 10
e8 10
e9 11
e10 11
e11 6
ymax
6
12
12
13
13
10
8
8
4
4
13
e6
10
e3
1/m
-2/5
1/3
-2/5
0
-4/3
-1/2
2
3/8
-3/4
2/3
e5
e4
e7
e8
5
e2
e9
e11
e10
0
0
5
10
15
Running the
Algorithm
y=0
SCB 1011
1/4
e10
4 -3/4
13
e6

10
e3
e9
1111
3/8
8
3/8
e5
e4
e7
e8

5
e2
e11
e10
0
0
April 8, 2015
e9
5
10
15
12
Running the
Algorithm
y=1
SLB 1 3/5
2
e2
6 -2/5
13
e6

10
6 2/3
6
e11
4
2/3
e3
e5
e4
e7
e8

5
10
9 1/2
1/4
e10
4 -3/4
e2

e11
e10
0
6/8
11 3/8
April 8, 2015
e9
8
3/8

e9
0
5
10
15
13
Running the
Algorithm
y=2
SLB 1 3/5
1/5
e2
6 -2/5
13
e6

10
7
6 1/3
2/3
e11
4
2/3
e3
e5
e4
e7
e8

5
8 1/2
3/4
9
e10
4 -3/4
e2

e11
e10
0
12
1/8
11 6/8
April 8, 2015
e9
8
3/8

e9
0
5
10
15
14
Running the
Algorithm
y=3
SLB 14/5
1/5
e2
6 -2/5
13
e6

10
78
1/3
e11
4
2/3
e3
e5
e4
e7
e8

5
8
8 3/4
e10
4 -3/4
e2

e11
e10
0
1/8
12 4/8
April 8, 2015
e9
8
3/8

e9
0
5
10
15
15
Running the
Algorithm
y=4
SLB
4/5
e2
6 -2/5
13
e6

10
8
e11
4
2/3
e3
e5
e4
e7
e8

5
8
e10
4 -3/4
e2

e11
e10
0
12 4/8
April 8, 2015
e9
8
3/8

e9
0
5
10
15
Remove these edges.
16
Running the
Algorithm
y=4
SLB
4/5
2/5
e2
6 -2/5
13
e6

10
12 7/8
4/8
e9
8
3/8
e3

e5
e4
e7
e8
5
e2
e11 and e10 are removed.
e11
e10
0
0
April 8, 2015
e9
5
10
15
17
Running the
Algorithm
y=5
SLB
2/5
0
e2
6 -2/5
13
e6

10
12 2/8
7/8
13
e9
8
3/8
e3

e5
e4
e7
e8
5
e2
e11
e10
0
0
April 8, 2015
e9
5
10
15
18
Running the
Algorithm
Remove this edge.
y=6
SLB
0
e2
6 -2/5
13
e6

10
910
1/2
e7
10 -1/2
e3

e5
e4
e7
e8
5
e8
12
10
8
2
e2

e11
e10
0
13 2/8
5/8
April 8, 2015
e9
8
3/8

e9
0
5
10
15
19
Add these edges.
1/3
e3
12 1/3

4
e4
12 -2/5

4
y=7
SLB 9 1/2
e5
13
0
e7
10 -1/2

8
2
13
e6
10
e3

5

0
e8
12
Running the
Algorithm
e2
13 5/8

e4
e7
e8
e9
e11
e10
0
e9
8
3/8
e5
5
10
15
Non-Simple Polygons
4/8/2015
College of Computer and Information Science, Northeastern University
21
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.
4/8/2015
College of Computer and Information Science, Northeastern University
22
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
4/8/2015
College of Computer and Information Science, Northeastern University
23
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
4/8/2015
College of Computer and Information Science, Northeastern University
24
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)
4/8/2015
College of Computer and Information Science, Northeastern University
25
Intuition
• intuition: reasonable definition of insideness 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)
4/8/2015
College of Computer and Information Science, Northeastern University
26
The Results Can be Different
4/8/2015
College of Computer and Information Science, Northeastern University
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