1. CS U540 Computer Graphics Prof. Harriet Fell Spring 2007
Lectures 16 – February 11, 2009
17 – February 12, 2009
18 – February 16, 2009
February 28, 2019

2. Clipping Lines G D C F A F’ G’ E B K H’ J H

3. Intersections We know how to find the intersections of a line segment
P + t(Q-P)
with the 4 boundaries
x = xmin
x = xmax
y = ymin
y = ymax Q P
February 28, 2019

4. Cohen-Sutherland Clipping
Assign a 4 bit outcode to each endpoint.
0000
1000
1100
0010
above left below right
0100
0110
1001
0001
0011
//above left below right
int outcode(x,y,xmin,xmax,ymin,ymax) {
int code=0;
if(x > xmax) code |= 1; //right
if(y < ymin) code |= 2; //below
if(x < xmin) code |= 4; //left
if(y > ymax) code |= 8; //above
return(code); }
February 28, 2019

5. Cohen-Sutherland Clipping
Identify lines that are trivially accepted or trivially rejected.
0000
1000
1100
0010
above left below right
0100
0110
1001
0001
0011
if(outcode(P)|outcode(Q)==0) accept
else if(outcode(P)&outcode(Q))!=0)
reject
else test further
February 28, 2019

6. Cohen-Sutherland continued
Clip against one boundary at a time, top, left, bottom, right.
Check for trivial accept or reject.
If a line segment PQ falls into the “test further” category then
if (outcode(P) & 1000  0)
replace P with PQ intersect y = top
else if (outcode(Q) & 1000  0)
replace Q with PQ intersect y = top
go on to next boundary
February 28, 2019

7. G G’
G’’
ACCEPT H’ H

8. Cohen-Sutherland Applet
By Patrick Min, CS Department, Princeton University
February 28, 2019

9. Liang-Barsky Clipping
Clip window interior is defined by:
xleft  x  xright
ybottom  y  ytop
February 28, 2019

10. Liang-Barsky continued
V1 = (x1, y1)
x = x0 + tx x = x1 - x0
y = y0 + ty y = y1 - y0
t = 0 at V0 t = 1 at V1
V0 = (x0, y0)
February 28, 2019

11. Liang-Barsky continued
Put the parametric equations into the inequalities:
xleft  x0 + tx  xright
ybottom  y0 + ty  ytop
-tx  x0 - xleft tx  xright - x0
-ty  y0 - ybottom ty  ytop - y0
These describe the interior of the clip window in terms of t.
February 28, 2019

12. Liang-Barsky continued
-tx  x0 - xleft tx  xright - x0
-ty  y0 - ybottom ty  ytop - y0
These are all of the form
tp  q
For each boundary, we decide whether to accept, reject, or which point to change depending on the sign of p and the value of t at the intersection of the line with the boundary.
February 28, 2019

13. x = xleft
p = - x V1
t 
t (-x)  (x0 – xleft)
t = (x0 – xleft)/(x0 – x1 )
is between 0 and 1.
x = x1 – x0 > 0 so p < 0
 replace V0 V0 V1
t = (x0 – xleft)/(x0 – x1 )
is between 0 and 1.
x = x1 – x0 < 0 so p > 0  replace V1
 t V0

14. x = tleft
p = - x V1
p < 0 so might replace V0
but
t = (x0 – xleft)/(x0 – x1 ) < 0
so no change. V0
t  V1 V0
p > 0
t > 1 V1 V0
t 
p < 0 so might replace V0
but
t = (x0 – xleft)/(x0 – x1 ) > 1
so reject.
 t V0 V1
p > 0 so might replace V1
but
t = (x0 – xleft)/(x0 – x1 ) < 0
so reject.

15. Liang-Barsky Rules 0 < t < 1, p < 0 replace V0
t < 0, p < 0 no change
t < 0, p > 0 reject
t > 1, p > 0 no change
t > 1, p < 0 reject
February 28, 2019

16. Polygon Clipping
February 28, 2019

17. Sutherland-Hodgeman Polygon Clipping Algorithm
Clip one boundary at a time: left, top, right, bottom.
Check each adjacent pair of vertices (P,Q), in order to make a new vertex list.
If P is out and Q is in, add intersection point with boundary and Q.
If P and Q are in, add Q.
If P is in and Q is out, add the intersection point with boundary only.
If P and Q are both out, add nothing.
February 28, 2019

18. Sutherland-Hodgeman Clipping ExampleV1
New Vertex List V2
Clip Left V2 V3 L2
If P is out and Q is in, add intersection point with boundary and Q.
If P is in and Q is out, add the intersection point with boundary only. V4
If P and Q are in, add Q. V6 L1 V5 L1 V5 L2 V3 V1 V4
February 28, 2019

19. Sutherland-Hodgeman Clipping ExampleV1
New Vertex List V2
Clip Left V2
Clip Top V3 L2 V4 V5 L1 L1 V5 L2 V3 V1 V4
February 28, 2019

20. Sutherland-Hodgeman Clipping ExampleV1
New Vertex List V2
Clip Top T2 T1 T1 V3 L2
If P is in and Q is out, add the intersection point with boundary only.
If P is out and Q is in, add intersection point with boundary and Q. V4
If P and Q are both out, add nothing.
If P and Q are in, add Q. V5 L1 L1 V5 L2 V3 T2 V4
February 28, 2019

21. Sutherland-Hodgeman Clipping Example
New Vertex List
Clip Right
Clip Top T2 T1 T1 V3 L2 V4 V5 L1 L1 V5 L2 V3 T2 V4
February 28, 2019

22. Sutherland-Hodgeman Clipping Example
New Vertex List
Clip Right T2 T1 R1 R2 R1 L2 V4 V5 L1 L1 V5 L2 V3 T2 R2 T1 V4
February 28, 2019

23. Sutherland-Hodgeman Clipping Example
New Vertex List T2 T1 R2 B1 R1 L2 B2 V5 L1 L1 V5 L2 T2 R2 T1
Clip Bottom B2 B1 R1 V4
February 28, 2019

24. Sutherland-Hodgeman Clipping Example
New Vertex List T2 T1 R2 B1 R1 L2 B2 V5 L1 L1 V5 L2 T2 R2 T1 B2 B1 R1
February 28, 2019

25. Sutherland-Hodgeman Exercise 1
Vertex List
initial after after after after
left top right bottom A B C D A D B C
February 28, 2019

26. Sutherland-Hodgeman Exercise 2
Vertex List
initial after after after after
left top right bottom H
I I
A A A
B B B
C C C
D D D
E E E
F F F
G G G
H H I A G R1 I D R4 B F R3 E R2 H C
February 28, 2019

27. Sutherland-Hodgeman Exercise 2
Vertex List
initial after after after after
left top right bottom H
I I
A A A R1 K
B B B R2 B1
C C C D D
D D D R3 B2
E E E R4 K
F F F G R4
G G G H G
H H I B3
I A B4 I A R1 A G R1 I D R4 B F B4 B3 B2 K R3 B1 E R2 H C
February 28, 2019