1. Computer Graphics
Clipping

2. Clipping
Analytically calculating the portions of primitives within the viewport

3. Why Clip?
Bad idea to rasterize outside of framebuffer bounds Also, don’t waste time scan converting pixels outside window

4. Clipping The naïve approach to clipping lines:
for each line segment
for each edge of viewport
find intersection point
pick “nearest” point
if anything is left, draw it
What do we mean by “nearest”?
How can we optimize this? A B C D

5. Trivial Accepts
Big optimization: trivial accept/rejects How can we quickly determine whether a line segment is entirely inside the viewport? A: test both endpoints.

6. Trivial Rejects
How can we know a line is outside viewport? A: if both endpoints on wrong side of same edge, can trivially reject line

7. Clipping Lines To Viewport
Combining trivial accepts/rejects
Trivially accept lines with both endpoints inside all edges of the viewport
Trivially reject lines with both endpoints outside the same edge of the viewport
Otherwise, reduce to trivial cases by splitting into two segments

8. Cohen-Sutherland Line Clipping
Divide viewplane into regions defined by viewport edges Assign each region a 4-bit outcode:
1001
1000
1010
0001
0000
0010
0101
0100
0110

9. Cohen-Sutherland Line Clipping
Assign an outcode to each vertex of line to test
Bit 1 = sign bit of (ymax – y)
If both outcodes = 0, trivial accept
bitwise OR
bitwise AND vertex outcodes together
if result  0, trivial reject

10. Cohen-Sutherland Line Clipping
If line cannot be trivially accepted or rejected, subdivide so that one or both segments can be discarded Pick an edge that the line crosses (how?) Intersect line with edge (how?) Discard portion on wrong side of edge and assign outcode to new vertex Apply trivial accept/reject tests; repeat if necessary

11. Cohen-Sutherland Line Clipping
If line cannot be trivially accepted or rejected, subdivide so that one or both segments can be discarded
Pick an edge that the line crosses
Check against edges in same order each time
For example: top, bottom, right, left E D C B A

12. Cohen-Sutherland Line Clipping
Intersect line with edge (how?) A B D E C

13. Cohen-Sutherland Line Clipping
Discard portion on wrong side of edge and assign outcode to new vertex Apply trivial accept/reject tests and repeat if necessary D C B A

14. Viewport Intersection Code
(x1, y1), (x2, y2) intersect with vertical edge at xright
yintersect = y1 + m(xright – x1), m=(y2-y1)/(x2-x1)
(x1, y1), (x2, y2) intersect with horizontal edge at ybottom
xintersect = x1 + (ybottom – y1)/m, m=(y2-y1)/(x2-x1)

15. Cohen-Sutherland Review
Use opcodes to quickly eliminate/include lines
Best algorithm when trivial accepts/rejects are common
Must compute viewport clipping of remaining lines
Non-trivial clipping cost
Redundant clipping of some lines
More efficient algorithms exist

16. Clipping Polygons
Clipping polygons is more complex than clipping the individual lines
Input: polygon
Output: original polygon, new polygon, or nothing
When can we trivially accept/reject a polygon as opposed to the line segments that make up the polygon?

17. How many sides can a clipped triangle have?
Why Is Clipping Hard?
What happens to a triangle during clipping? Possible outcomes:
triangle  quad
triangle  triangle
triangle  5-gon
How many sides can a clipped triangle have?

18. How many sides?
Seven…

19. Why Is Clipping Hard?
A really tough case:

20. Why Is Clipping Hard? A really tough case:
concave polygon  multiple polygons

21. Sutherland-Hodgeman Clipping
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the viewport edge’s equation

22. Sutherland-Hodgeman Clipping
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all edges, the polygon is fully clipped

23. Sutherland-Hodgeman Clipping
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all edges, the polygon is fully clipped

24. Sutherland-Hodgeman Clipping
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all edges, the polygon is fully clipped

25. Sutherland-Hodgeman Clipping
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all edges, the polygon is fully clipped

26. Sutherland-Hodgeman Clipping
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all edges, the polygon is fully clipped

27. Sutherland-Hodgeman Clipping
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all edges, the polygon is fully clipped

28. Sutherland-Hodgeman Clipping
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all edges, the polygon is fully clipped

29. Sutherland-Hodgeman Clipping
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all edges, the polygon is fully clipped

30. Sutherland-Hodgeman Clipping: The Algorithm
Basic idea:
Consider each edge of the viewport individually
Clip the polygon against the edge equation
After doing all edges, the polygon is fully clipped

31. Sutherland-Hodgeman Clipping
Input/output for algorithm:
Input: list of polygon vertices in order
Output: list of clipped poygon vertices consisting of old vertices (maybe) and new vertices (maybe)
Note: this is exactly what we expect from the clipping operation against each edge

32. Sutherland-Hodgeman Clipping
Sutherland-Hodgman basic routine:
Go around polygon one vertex at a time
Current vertex has position p
Previous vertex had position s, and it has been added to the output if appropriate

33. Sutherland-Hodgeman Clipping
Edge from s to p takes one of four cases:
(Orange line can be a line or a plane)
inside
outside s p
p output
inside
outside s p
i output
inside
outside s p
no output
inside
outside s p
i output p output

34. Sutherland-Hodgeman Clipping
Four cases:
s inside plane and p inside plane
Add p to output
Note: s has already been added
s inside plane and p outside plane
Find intersection point i
Add i to output
s outside plane and p outside plane
Add nothing
s outside plane and p inside plane
Add i to output, followed by p