1. Joshua Barczak CMSC 435 UMBC
Clipping
Joshua Barczak
CMSC 435
UMBC

2. Object-Order Rendering
We are here
Object-Space
Primitives
World-space Primitives
Camera-Space Primitives
Clip-Space Primitives
Clip
Modeling
XForm
Viewing
XForm
Projection
XForm
Clip-Space Primitives
Rasterize
Window
XForm
Viewport
XForm
Displayed Pixels
Pixels
Raster-Space Primitives

3. Viewing Transform The job of the viewing transform is:
Get us from these axes
Onto these axes

4. Projection Transform The job of the projection transform is:
Cram the visible universe into a unit box
Project

5. Canonical View Volume (GL)
(-1,1,1)
(1,1,1)
Top Edge of Screen
(0,1,-1)
Center of screen
Left Edge of Screen
(0,0,-1)
(-1,0,-1)
(1,0,-1)
(1,-1,1)
Points outside the cube are not visible…
(1,-1,-1)
(0,-1,-1)
D3D uses z=0. I like theirs better…
Bottom Edge of Screen

6. Primitive States Invisible Partially Visible Visible Our Reaction:
Trivial Reject:
Throw it away, it does not exist…
Remove the parts that do not exist…
Trivial Accept:
Pass it through

7. Definitions Culling: Clipping Discarding invisible primitives
Removing invisible parts of primitives

8. Basic Idea For each primitive: Transform to C.V.V. (clip space)
Cull test: Is it completely outside?
Trivial Reject
Else: Is it completely inside?
Trivial Accept
Else
Clip it

9. Why do we do this? Culling Clipping Efficiency
Simplicity of rasterization
No boundary cases
Efficiency of rasterization
Skip invisible pixels
Stability of rasterization
Precision

10. Line Clipping
Turns lines into other lines

11. Line Clipping(3D) Same thing, but edges are planes now…
Same algorithms, different math
2D is easier to draw…

12. Line Clipping
Point is in clip box if:
xmax,ymax
xmin,ymin

13. Line Clipping Endpoints in box?? Both endpoints inside Trivial accept
One endpoint inside
Will clip
Both endpoints outside
Don’t know yet…??

14. Line Clipping Region Check?
Trivial reject if both endpoints outside the same boundary…
max(x0,x1) <= xmin min(x0,x1) >= xmax min(y0,y1) <= ymin max(y0,y1) >= ymax

15. Cohen-Sutherland Line Clipping
Each region assigned a 4-bit code (outcode)
1st bit – above top
y > ymax
2nd bit – below bottom
y < ymin
3rd bit – right of right
x > xmax
4th bit – left of left
x < xmin

16. Cohen-Sutherland Line Clipping
Efficient bitcode calculation:
First bit is the sign bit of ymax – y
Second bit is y – ymin
Third bit is the sign bit of xmax – x
Forth bit is x – xmin

17. Cohen-Sutherland Line Clipping
Trivial accept if both endpoints zero
code1|code2 ==0
Trivial reject if both endpoints ‘outside’ same edge
code1 & code2 != 0
Otherwise, need to split the line

18. Cohen-Sutherland Line Clipping
code1 = outcode from endpoint1
code2 = outcode from endpoint2
if (code1 | code2 == 0) then
trivial_accept
else if (code1 & code2 != 0) then
trivial_reject
else
clip against left
clip against right
clip against bottom
clip against top
if (anything is left) then
accept clipped segment

19. Cohen-Sutherland Line Clipping
Clipping to an edge:
If one endpoint sticks out, move it in
Move one coordinate
Re-compute the other
At most one point is outside, due to region check

20. Cohen-Sutherland Line Clipping
Update and re-check outcodes after clipping…
Needs clip
Trivial Reject

21. Liang-Barsky Line Clipping
First: Trivial Reject based on AABB:
NOT < or >. This is important…

22. Liang-Barsky Line Clipping
Optional: Trivial Accept based on AABB:

23. Liang-Barsky Line Clipping
T=1
Next: Ray-box intersection
tymax
txmax
tymin
T=0
txmin
…Reject if t0 >= t1

24. Liang-Barsky Line Clipping
The divides are fine. Don’t panic…
We know that:
xmin
xmax y0 y1
… or the trivial reject test passes
IEEE rules handle division of non-zero by zero Nans are evil, but infinities are not…

25. Triangle Culling Trivial cases All vertices outside same box edge
All vertices inside box
Box corners outside triangle edge

26. Triangle Culling The edge test is 3 point-line tests (not 12)
Choose corner based on edge normal:
Nx > 0 ? xmin : xmax
Ny > 0 ? ymin : ymax

27. Need to factor these into all your tests…
Triangle Culling
Nasty edge cases
Edge in plane
Vert in plane
Entire triangle in plane
Need to factor these into all your tests…

28. Triangle Clipping General case…
There will be two vertices on one side, one on the other side

29. Triangle Clipping General case…
Compute edge/plane intersection locations
Two line clips…

30. Triangle Clipping General case…
If “the one” is inside, keep the triangle

31. Triangle Clipping General case…
If “the one” is outside, create two triangles
Pass both to subsequent clip tests
Gets expensive very quickly…

32. Triangle Clipping: Example

33. Triangle Clipping: Example

34. Triangle Clipping: Example

35. Triangle Clipping: Example

36. Triangle Clipping: Example

37. Guard Bands Add padding to turn clips into trivial reject/accept
Small triangles less likely to need clipping
Which is good, because they’re more numerous
Guard band
Viewport

38. Sutherland-Hodgeman Polygon Clipping
For each edge, walk along vertex loop
Add 0, 1, or 2 vertices to output

39. Sutherland-Hodgman Polygon Clipping
At each step, 1 of 4 possible cases arises
1) Edge is completely inside clip boundary, so add vertex p to the output list
2) Intersection i is output as vertex because it intersects with boundary
3) Both vertices are outside boundary, so neither is output
4) Intersection i and vertex p both added to output list 39 39

40. Homogeneous Clipping
The projection transform does strange and wonderful things…
Recall:
For perspective projection, typically w’=z

41. Homogeneous Clipping
The following hold for all points in the view frustum
Assuming sane view/projection matrices
OpenGL spec is undefined for w<0

42. Projection Matrix Behavior
Near plane
Far plane
Eye
z=w
w=0
z=-w

43. Homogeneous Clipping
Can clip/cull to w planes on each axis, prior to division
Avoids division for culled primitives
But wait, it gets weirder…! w
x=-w
x=w -x x

44. Homogeneous Clipping
Blinn, 1978
Curiouser and curiouser…

45. Homogeneous Clipping Motion of projected point, from P1 to P2
Blinn, 1978
P2 is behind the eye….

46. Homogeneous Clipping
Tooo Infinity! And beyond!
Blinn, 1978

47. Homogeneous Clipping
This happens for triangles too…
Olano and Greer

48. Homogeneous Clipping
This is what we want… w
x=-w
x=w
w=1 -x x

49. Homogeneous Clipping
…but this is what we’ll get… w
x=-w
x=w
w=1 -x x

50. Homogeneous Clipping …and this is what the user will think they saw… w
x=-w
x=w
w=1 -x x

51. Homogeneous Clipping Worse yet…
Invisible segments magically become visible… w
x=-w
x=w
w=1 -x x

52. Why do we do this? Culling Clipping Efficiency
Simplicity of rasterization
No boundary cases
Efficiency of rasterization
Skip invisible pixels
Stability of rasterization
Precision
All of these are reasons we might want to clip. But…

53. The Real Reason We Clip:
Correct perspective for points behind the eye
If you project without clipping, you will draw the wrong line!
You cannot detect this post-projection w
x=-w
x=w
w=1 -x x
Blue and red lines have same projected endpoints

54. Homogeneous Clipping
Without clip (WRONG) w
x=-w
x=w
w=1 -x x

55. Homogeneous Clipping Trivial Reject cases: w x=-w x=w w=1 Outside x=w
Outside Both

56. Homogeneous Clipping Replace x=xmin/xmax with x=+/-w Ditto for y and z
Use clipped ‘w’ values in subsequent tests w
x=-w
x=w
w=1 -x x

57. Homogeneous Clipping w
x=-w
x=w
w=1 -x x