1. (c) 2002 University of Wisconsin, CS559
Last Time
Finished viewing: Now you know how to:
Define a region of space that you wish to view – the view volume
Transform it into a generic view volume – canonical view volume
Using orthographic or perspective projection
Very brief intro to clipping
02/28/02
(c) 2002 University of Wisconsin, CS559

2. (c) 2002 University of Wisconsin, CS559
Today
Clipping in all its detail
02/28/02
(c) 2002 University of Wisconsin, CS559

3. Sutherland-Hodgman Clip
Clip the polygon against each edge of the clip region in turn
Clip polygon each time to line containing edge
Only works for convex clip regions (Why?)
02/28/02
(c) 2002 University of Wisconsin, CS559

4. Sutherland-Hodgman Clip
To clip a polygon to a line/plane:
Consider the polygon as a list of vertices
One side of the line/plane is considered inside the clip region, the other side is outside
We are going to rewrite the polygon one vertex at a time – the rewritten polygon will be the polygon clipped to the line/plane
Check start vertex: if “inside”, emit it, otherwise ignore it
Continue processing vertices as follows…
02/28/02
(c) 2002 University of Wisconsin, CS559

5. Sutherland-Hodgman (3)
Look at the next vertex in the list, and the edge from the last vertex to the next. If the
polygon edge crosses the clip line/plane going from out to in: emit crossing point, next vertex
polygon edge crosses clip line/plane going from in to out: emit crossing
polygon edge goes from out to out: emit nothing
polygon edge goes from in to in: emit next vertex
02/28/02
(c) 2002 University of Wisconsin, CS559

6. Sutherland-Hodgman (4)
Inside
Outside
Inside
Outside
Inside
Outside
Inside
Outside s p i s p p s i p s
Output p
Output i
No output
Output i and p
02/28/02
(c) 2002 University of Wisconsin, CS559

7. Inside-Outside Testing
Lines/planes store a vector pointing toward the outside of the clip region – the outward pointing normal
Could re-define for inward pointing
Dot products give inside/outside information
Note that x is any point on the clip line/plane
Outside
Inside x n f i s
02/28/02
(c) 2002 University of Wisconsin, CS559

8. Finding Intersection Pts
Use the parametric form for the edge between two points, x1 and x2:
For planes of the form x=a:
Similar forms for y=a, z=a
Solution for general plane can also be found
02/28/02
(c) 2002 University of Wisconsin, CS559

9. Inside/Outside in Screen Space
In canonical screen space, clip planes are xs=±1, ys=±1, zs=±1
Inside/Outside reduces to comparisons before perspective divide
02/28/02
(c) 2002 University of Wisconsin, CS559

10. Hardware Sutherland-Hodgman
Suitable for hardware implementation
Only need the clip edge, the endpoints of the current edge, and the last output point
Polygon edges are output as they are found, and passed right on to the next clip region edge
Clip Top
Clip Right
Clip Bottom
Vertices in
Clip Far
Clip Near
Clip Left
Clipped vertices out
02/28/02
(c) 2002 University of Wisconsin, CS559

11. (c) 2002 University of Wisconsin, CS559
Other Ways to Reject
If a polygonal object is closed, then no back-facing face is visible
Front-facing faces must occlude all back-facing ones
Reject back-facing polygons in view space
Transform face normal and check
OpenGL supports optional back-face culling (and front-face culling too)
Bounding volumes enclosing many polygons can be checked against the view volume
Done in software in world or view space
Visibility can reject whole chunks of geometry without even looking at them
02/28/02
(c) 2002 University of Wisconsin, CS559

12. (c) 2002 University of Wisconsin, CS559
Clipping In General
Apart from clipping to the view volume, clipping is a basic operation in many other algorithms
Breaking space up into chunks
2D drawing and windowing
Modeling
May require more complex geometry than rectangular boxes
02/28/02
(c) 2002 University of Wisconsin, CS559

13. Additional Clipping Planes
Useful for doing things like cut-away views
Use a clip plane to cut off part of the object
Only works if piece to be left behind is convex
OpenGL allows you to do it
Also one way to use OpenGL to identify objects in a region of space (uses the selection mechanism)
02/28/02
(c) 2002 University of Wisconsin, CS559

14. (c) 2002 University of Wisconsin, CS559
Clipping Lines
Lines can also be clipped by Sutherland-Hodgman
Slower than necessary, unless you already have hardware
Better algorithms exist
Cohen-Sutherland
Liang-Barsky
Nicholl-Lee-Nicholl (we won’t cover this one – only good for 2D)
02/28/02
(c) 2002 University of Wisconsin, CS559

15. (c) 2002 University of Wisconsin, CS559
Cohen-Sutherland (1)
Works basically the same as Sutherland-Hodgman
Was developed earlier
Clip line against each edge of clip region in turn
If both endpoints outside, discard line and stop
If both endpoints in, continue to next edge (or finish)
If one in, one out, chop line at crossing pt and continue
Works in both 2D and 3D for convex clipping regions
02/28/02
(c) 2002 University of Wisconsin, CS559

16. (c) 2002 University of Wisconsin, CS559
Cohen-Sutherland (2) 1 2 1 2 3 3 4 4 3 3 4 4 1 2 1 2
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(c) 2002 University of Wisconsin, CS559

17. (c) 2002 University of Wisconsin, CS559
Cohen-Sutherland (3)
Some cases lead to premature acceptance or rejection
If both endpoints are inside all edges
If both endpoints are outside one edge
General rule of clipping – if a fast test can cover many cases, do it first
02/28/02
(c) 2002 University of Wisconsin, CS559

18. Cohen-Sutherland - Details
Only need to clip line against edges where one endpoint is out
Use outcode to record endpoint in/out wrt each edge. One bit per edge, 1 if out, 0 if in.
Trivial reject:
outcode(x1)&outcode(x2)!=0
Trivial accept:
outcode(x1)|outcode(x2)==0
Which edges to clip against?
outcode(x1)^outcode(x2) 1 2
0010 3 4
0101
02/28/02
(c) 2002 University of Wisconsin, CS559

19. Liang-Barsky Clipping
Parametric clipping - view line in parametric form and reason about the parameter values
More efficient, as not computing the coordinate values at irrelevant vertices
Works for rectilinear clip regions in 2D or 3D
Clipping conditions on parameter: Line is inside clip region for values of t such that (for 2D):
02/28/02
(c) 2002 University of Wisconsin, CS559

20. (c) 2002 University of Wisconsin, CS559
Liang-Barsky (2)
Infinite line intersects clip region edges when:
where
Note: Left edge is 1, right edge is 2, top edge is 3, bottom is 4
02/28/02
(c) 2002 University of Wisconsin, CS559

21. (c) 2002 University of Wisconsin, CS559
Liang-Barsky (3)
When pk<0, as t increases line goes from outside to inside - entering
When pk>0, line goes from inside to outside – leaving
When pk=0, line is parallel to an edge (clipping is easy)
If there is a segment of the line inside the clip region, sequence of infinite line intersections must go: enter, enter, leave, leave
02/28/02
(c) 2002 University of Wisconsin, CS559

22. (c) 2002 University of Wisconsin, CS559
Liang-Barsky (4)
Leave
Enter
Leave
Leave
Leave
Enter
Enter
Enter
02/28/02
(c) 2002 University of Wisconsin, CS559

23. Liang-Barsky - Algorithm
Compute entering t values, which are qk/pk for each pk<0
Compute leaving t values, which are qk/pk for each pk>0
Parameter value for small t end of line is:tsmall= max(0, entering t’s)
parameter value for large t end of line is: tlarge=min(1, leaving t’s)
if tsmall<tlarge, there is a line segment - compute endpoints by substituting t values
Improvement (and actual Liang-Barsky):
compute t’s for each edge in turn (some rejects occur earlier like this)
02/28/02
(c) 2002 University of Wisconsin, CS559

24. Weiler Atherton Polygon Clipping
Faster than Sutherland-Hodgman for complex polygons
For clockwise polygon:
for out-to-in pair, follow usual rule
for in-to-out pair, follow clip edge
Easiest to start outside
02/28/02
(c) 2002 University of Wisconsin, CS559

25. (c) 2002 University of Wisconsin, CS559
General Clipping
Clipping general against general polygons is quite hard
Outline of Weiler algorithm:
Replace crossing points with vertices
Double all edges and form linked lists of edges
Change links at vertices
Enumerate polygon patches
Can use clipping to break concave polygon into convex pieces; main issue is inside-outside for edges
02/28/02
(c) 2002 University of Wisconsin, CS559

26. (c) 2002 University of Wisconsin, CS559
Weiler Algorithm (1)
02/28/02
(c) 2002 University of Wisconsin, CS559

27. (c) 2002 University of Wisconsin, CS559
Rearranging pointers
makes it possible to
enumerate all components
of the intersection C h a n g e s t o
02/28/02
(c) 2002 University of Wisconsin, CS559

28. (c) 2002 University of Wisconsin, CS559
Where We Stand
At this point we know how to:
Convert points from local to screen coordinates
Clip polygons and lines to the view volume
Next thing:
Determine which pixels are covered by any given point, line or polygon
02/28/02
(c) 2002 University of Wisconsin, CS559

29. (c) 2002 University of Wisconsin, CS559
Drawing Points
When points are mapped into window coordinates, they could land anywhere – not just at a pixel center
Solution is the simple, obvious one
Map to window space
Fill the closest pixel
Can also specify a radius – fill a square of that size, or fill a circle
Square is faster
02/28/02
(c) 2002 University of Wisconsin, CS559

30. (c) 2002 University of Wisconsin, CS559
Drawing Lines
Task: Decide which pixels to fill (samples to use) to represent a line
We know that all of the line lies inside the visible region (clipping gave us this!)
Issues:
If slope between -1 and 1, one pixel per column. Otherwise, one pixel per row
Constant brightness? Lines of the same length should light the same number of pixels (we normally ignore this)
Anti-aliasing? (Getting rid of the “jaggies”)
02/28/02
(c) 2002 University of Wisconsin, CS559

31. Line Drawing Algorithms
Consider lines of the form y=m x + c, where m=y/x, 0<m<1, integer coordinates
All others follow by symmetry
Variety of slow algorithms (Why slow?):
step x, compute new y at each step by equation, rounding:
step x, compute new y at each step by adding m to old y, rounding:
02/28/02
(c) 2002 University of Wisconsin, CS559

32. Bresenham’s Algorithm Overview
Plot the pixel whose y-value is closest to the line
Given (xi,yi), must choose from either (xi+1,yi+1) or (xi+1,yi)
Idea: compute a decision variable
Value that will determine which pixel to draw
Easy to update from one pixel to the next
02/28/02
(c) 2002 University of Wisconsin, CS559

33. (c) 2002 University of Wisconsin, CS559
Decision Variable
Decision variable is:
yi+1 d2 d1 yi xi
xi+1
02/28/02
(c) 2002 University of Wisconsin, CS559

34. (c) 2002 University of Wisconsin, CS559
What Can We Decide?
d1<d2 => pi negative => next point at (xi+1,yi)
d1>d2 => pi positive => next point at (xi+1,yi+1)
So, we know what to draw based on the decision variable
How do we update it? What is pk+1?
02/28/02
(c) 2002 University of Wisconsin, CS559

35. Updating The Decision Variable
If yi+1=yi+1:
If yi+1=yi:
What is p1 (assuming integer endpoints)?
02/28/02
(c) 2002 University of Wisconsin, CS559

36. Bresenham’s Algorithm
For integers, slope between 0 and 1:
x=x1, y=y1, p=2 dy - dx, draw (x, y)
until x=x2
x=x+1
p>0 ? y=y+1, draw (x, y), p=p+2 y - 2 x
p<0? y=y, draw (x, y), p=p+2 y
Compute the constants once at the start
Only does add and comparisons
Floating point has slightly more difficult initialization
02/28/02
(c) 2002 University of Wisconsin, CS559

37. (c) 2002 University of Wisconsin, CS559
Example: (2,2) to (7,6)
x=5, y=4
i x y p 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8
02/28/02
(c) 2002 University of Wisconsin, CS559