1. (c) 2002 University of Wisconsin, CS559
Last Time
Drawing lines
Inside/Outside tests for polygons
10/22/02
(c) 2002 University of Wisconsin, CS559

2. (c) 2002 University of Wisconsin, CS559
Today
Drawing Polygons
10/22/02
(c) 2002 University of Wisconsin, CS559

3. (c) 2002 University of Wisconsin, CS559
What is inside - 1?
Easy for simple polygons - no self intersections or holes
OpenGL requires these. Undefined for other cases
OpenGL also requires convex polygons
For general polygons, three rules are possible:
Non-exterior rule: A point is inside if every ray to infinity intersects the polygon
Non-zero winding number rule: Draw a ray to infinity that does not hit a vertex, if the number of edges crossing in one direction is not equal to the number crossing the other way, the point is inside
Parity rule: Draw a ray to infinity and count the number or edges that cross it. If even, the point is outside, if odd, it’s inside
10/22/02
(c) 2002 University of Wisconsin, CS559

4. (c) 2002 University of Wisconsin, CS559
Inside/Outside Rules
Polygon
Non-exterior
Non-zero Winding No.
Parity
10/22/02
(c) 2002 University of Wisconsin, CS559

5. (c) 2002 University of Wisconsin, CS559
What is inside - 2?
Assume sampling with an array of spikes
If spike is inside, pixel is inside
10/22/02
(c) 2002 University of Wisconsin, CS559

6. (c) 2002 University of Wisconsin, CS559
What is inside - 2?
Assume sampling with an array of spikes
If spike is inside, pixel is inside
10/22/02
(c) 2002 University of Wisconsin, CS559

7. (c) 2002 University of Wisconsin, CS559
Ambiguous Case
Ambiguous cases: What if a pixel lies on an edge?
Problem because if two polygons share a common edge, we don’t want pixels on the edge to belong to both
Ambiguity would lead to different results if the drawing order were different
Rule: if (x+, y+) is in, (x,y) is in
What if a pixel is on a vertex? Does our rule still work?
10/22/02
(c) 2002 University of Wisconsin, CS559

8. (c) 2002 University of Wisconsin, CS559
Ambiguous Case 1
Rule:
On edge? If (x+, y+) is in, pixel is in
Which pixels are colored?
10/22/02
(c) 2002 University of Wisconsin, CS559

9. (c) 2002 University of Wisconsin, CS559
Ambiguous Case 1
Rule:
Keep left and bottom edges
Assuming y increases in the up direction
If rectangles meet at an edge, how often is the edge pixel drawn?
10/22/02
(c) 2002 University of Wisconsin, CS559

10. (c) 2002 University of Wisconsin, CS559
Ambiguous Case 2
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(c) 2002 University of Wisconsin, CS559

11. (c) 2002 University of Wisconsin, CS559
Ambiguous Case 2 ? ? ? or ?
10/22/02
(c) 2002 University of Wisconsin, CS559

12. (c) 2002 University of Wisconsin, CS559
Really Ambiguous
We will accept ambiguity in such cases
The center pixel may end up colored by one of two polygons in this case
Which two? 1 6 2 5 3 4
10/22/02
(c) 2002 University of Wisconsin, CS559

13. (c) 2002 University of Wisconsin, CS559
Exploiting Coherence
When filling a polygon
Several contiguous pixels along a row tend to be in the polygon - a span of pixels
Scanline coherence
Consider whole spans, not individual pixels
The pixels required don’t vary much from one span to the next
Edge coherence
Incrementally update the span endpoints
10/22/02
(c) 2002 University of Wisconsin, CS559

14. (c) 2002 University of Wisconsin, CS559
Sweep Fill Algorithms
Algorithmic issues:
Reduce to filling many spans
Which edges define the span of pixels to fill?
How do you update these edges when moving from span to span?
What happens when you cross a vertex?
10/22/02
(c) 2002 University of Wisconsin, CS559

15. (c) 2002 University of Wisconsin, CS559
Spans
Process - fill the bottom horizontal span of pixels; move up and keep filling
Have xmin, xmax for each span
Define:
floor(x): largest integer < x
ceiling(x): smallest integer >=x
Fill from ceiling(xmin) up to floor(xmax)
Consistent with convention
10/22/02
(c) 2002 University of Wisconsin, CS559

16. (c) 2002 University of Wisconsin, CS559
Algorithm
For each row in the polygon:
Throw away irrelevant edges
Obtain newly relevant edges
Fill span
Update current edges
Issues:
How do we update existing edges?
When is an edge relevant/irrelevant?
All can be resolved by referring to our convention about what polygon pixel belongs to
10/22/02
(c) 2002 University of Wisconsin, CS559

17. (c) 2002 University of Wisconsin, CS559
Updating Edges
Each edge is a line of the form:
Next row is:
So, each current edge can have it’s x position updated by adding a constant stored with the edge
Other values may also be updated, such as depth or color information
10/22/02
(c) 2002 University of Wisconsin, CS559

18. When are Edges Relevant (1)
Use figures and convention to determine when edge is irrelevant
For y<ymin and y>=ymax of edge
Similarly, edge is relevant when y>=ymin and y<ymax of edge
What about horizontal edges?
m’ is infinite
10/22/02
(c) 2002 University of Wisconsin, CS559

19. When are Edges Relevant (2)
Convex polygon:
Always only two edges active 2
1,2 1
1,3 3
3,4 4
10/22/02
(c) 2002 University of Wisconsin, CS559

20. When are Edges Relevant (3)
Horizontal edges? 2 2? 1 3
1,3 4? 4
10/22/02
(c) 2002 University of Wisconsin, CS559

21. (c) 2002 University of Wisconsin, CS559
Sweep Fill Details
Maintain a list of active edges in case there are multiple spans of pixels - known as Active Edge List.
For each edge on the list, must know: x-value, maximum y value of edge, m’
Maybe also depth, color…
Keep all edges in a table, indexed by minimum y value - Edge Table
For row = min to row=max
AEL=append(AEL, ET(row));
remove edges whose ymax=row
sort AEL by x-value
fill spans
update each edge in AEL
10/22/02
(c) 2002 University of Wisconsin, CS559

22. (c) 2002 University of Wisconsin, CS559
Edge Table
Row: 6 6 5 5 4 4 6 4 3 3 2 2 1 2 4 6 6 1 6 6 1 2 3 4 5 6
ymax
xmin
1/m
10/22/02
(c) 2002 University of Wisconsin, CS559

23. Active Edge List (shown just before filling each row)6 6 5 4 6 6 6 5 4 4 6 6 6 4 3 2 4 6 6 3 2 2 4 6 6 2 1 2 4 6 6 1 6 6 1 2 3 4 5 6
ymax x
1/m
10/22/02
(c) 2002 University of Wisconsin, CS559

24. (c) 2002 University of Wisconsin, CS559
Edge Table
Row: 6 6 5 5 4 4 3 3 2 2 1 2 1 5 6 -1 5 1 6 6 1 2 3 4 5 6
ymax
xmin
1/m
10/22/02
(c) 2002 University of Wisconsin, CS559

25. (c) 2002 University of Wisconsin, CS559
Active Edge List
Row: 6 6 5 5 4 3 -1 5 5 1 5 4 3 4 1 5 4 -1 5 3 2 3 1 5 5 -1 5 2 1 2 1 5 6 -1 5 1 6 6 1 2 3 4 5 6
ymax x
1/m
10/22/02
(c) 2002 University of Wisconsin, CS559

26. (c) 2002 University of Wisconsin, CS559
Comments
Sort is quite fast, because AEL is usually almost in order
OpenGL limits to convex polygons, meaning two and only two elements in AEL at any time, and no sorting
Can generate memory addresses (for pixel writes) efficiently
Does not require floating point - next slide
10/22/02
(c) 2002 University of Wisconsin, CS559

27. Avoiding Floating Point
For edge, m=x/y, which is a rational number
View x as xi+xn/y, with xn<y. Store xi and xn
Then x->x+m’ is given by:
xn=xn+x
if (xn>=y) { xi=xi+1; xn=xn- y }
Advantages:
no floating point
can tell if x is an integer or not, and get floor(x) and ceiling(x) easily, for the span endpoints
10/22/02
(c) 2002 University of Wisconsin, CS559