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Clipping
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Culling and Clipping
•
What can’t we see?
- anything occluded by another object (HSR)
- anything outside view volume
•
Today: clipping
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5
6
7
8
9
10
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Clipping Against a Rectangular Region
Multiple Cases
B
F
E
C
A
Clip Rectangle
G
D
H
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Division of Space
ymax
Clip Region
ymin
x min
x max
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Cohen-Sutherland Clipping:
“Outcodes”
y  ymax
1
b0
x  x max
0
1
Example :
0
b1 b2 b3
y  ymin
x  xmin
1
b0  
0
y  ymax
y  ymax
1
b1  
0
y  ymin
y  ymin
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Cohen-Sutherland Clipping:
Region Outcodes
1001
1000
1010
0001
0000
0010
0101
0100
0110
ymax
ymin
x min
x max
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Cohen-Sutherland Clipping:
Trivial Acceptance: O(P0) = O(P1) = 0
1001
1000
ymax
1010
P1
0001
0000
0010
0100
0110
P0
ymin
0101
x min
x max
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Cohen-Sutherland Clipping:
Trivial Rejection: O(P0) & O(P1)  0
P0
1001
1000
1010
P1
ymax
P0
0001
ymin
0000
0010
P0
0101
x min
P1
P1
0100
0110
x max
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Cohen-Sutherland Clipping:
O(P0) =0 , O(P1)  0
1001
1000
1010
ymax
P0
0001
0000
P1
0010
P0
P0
ymin
0100
0101
0110
P1
x min
x max
P1
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Cohen-Sutherland Clipping: The Algorithm
• Compute the outcodes for the two vertices
• Test for trivial acceptance or rejection
• Select a vertex for which outcode is not zero
- There will always be one
• Select the first nonzero bit in the outcode to define the
boundary against which the line segment will be clipped
• Compute the intersection and replace the vertex with the
intersection point
• Compute the outcode for the new point and iterate
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Cohen-Sutherland Clipping:
Example 1
A
1001
1000
1010
B
ymax
C
0001
0000
0010
0101
0100
0110
ymin
x min
x max
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Cohen-Sutherland Clipping:
Example 1
1001
1000
1010
B
ymax
C
0001
0000
0010
0101
0100
0110
ymin
x min
x max
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Chopping at each boundary
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Cohen-Sutherland Clipping:
Example 2
1001
1000
1010
E
ymax
D
0001
0000
ymin
C
0010
B
A
0100
0101
x min
0110
x max
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Cohen-Sutherland Clipping:
Example 2
1001
1000
1010
E
ymax
D
0001
0000
ymin
C
0010
B
0100
0101
x min
0110
x max
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Cohen-Sutherland Clipping:
Example 2
1001
1000
1010
ymax
D
0001
0000
ymin
C
0010
B
0100
0101
x min
0110
x max
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Cohen-Sutherland Clipping:
Example 2
1001
1000
0001
0000
1010
ymax
ymin
C
0010
B
0100
0101
x min
0110
x max
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Cohen-Sutherland Clipping:
Advantages/Extension
• Easily extended to 3 dimensions by adding two bits to the
outcode for the z axis.
• Calculations then reduce to intersection of line with plane
• Algorithm most efficient when most segments can either be
trivially accepted or trivially rejected
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Cohen Sutherland in 3D
• Use 6-bit outcodes
• When needed, clip line segment against planes
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Other Clipping Algorithm
•
Cyrus–Beck, Liang–Barsky
-
generally, these try to be more clever about intersections
- represent lines parametrically, solve for intersection t
values
 P(t) = P0 + t(P1-P0)
- can clip in fewer steps than Cohen–Sutherland
- try to increase number of trivial rejection cases
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Parametric Representation of Lines
p1  p(1)
p 0  p(0)
p ( )
p( )  (1   )p 0  p1
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Liang-Barsky Clipping
• Consider the parametric form of a line segment
p() = (1-)p1+ p2 1    0
p2
p1
• We can distinguish between the cases by looking at the ordering of the
values of  where the line determined by the line segment crosses the lines
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Liang-Barsky Parametric Clipping
p( )  p 0   (p1  p 0 )
Edge Ei
Outside
Inside
p Ei

p1

N i  p( )  p Ei  0


N i  p( )  p Ei  0


N i  p( )  p Ei  0
p0
Ni
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Potentially Entering (PE) and Potentially Leaving
(PL) Points
Edge Ei
Outside
Edge Ei
Inside
Inside
Outside
p1
p1
p0
Ni
Potentially Entering
(PE) Point
Ni  p1  p 0   0
p0
Ni
Potentially Leaving
(PL) Point
Ni  p1  p0   0
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Liang-Barsky Clipping:
Computing the Intersection


N i  p( )  p Ei  0
p( )  p 0   (p1  p 0 )


N i  p 0  p Ei  N i  p1  p 0   0
Let D  (p1  p 0 ) and solve for  :
=

N i  p 0  p Ei
 Ni  D

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Liang-Barsky Clipping:
Potentially Leaving vs. Potentially Entering
p1
p1
PE
PL
ymax
p1
PL
PL
p0
PE
PL
p0
PE
PE
ymin
p0
x min
= “Inside”
x max
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Liang-Barsky Clipping:
Algorithm Strategy
• Find the largest PE greater than zero.
• Find the smallest PL less than one.
• Reject the segment if PE > PL.
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Liang-Barsky Clipping:
Pseudocode of Algorithm
for (each line segment to be clipped)
alpha_E=0; alpha_L=1;
for (each candidate intersection with a clip edge) {
if (Ni•D!=0) { /*edges not parallel to line*/
calculate alpha;
use sign of Ni•D to class alpha as PE
or PL;
if (PE) alpha_E = max(alpha_E,alpha);
if (PL) alpha_L = min(alpha_L,alpha);
}
}
if (alpha_E > alpha_L)
return NULL;
else
return P(alpha_E) and P(alpha_L) as clip
intersections
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Polygon Clipping:
Convex Polygons
5
4
5’
4’
1
3
1
3
2
2
Line segment clipping is done in order. New vertices
are generated at clip point. External vertices are
eliminated.
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Polygon Clipping:
The Convexity Problem
Single object becomes multiple objects.
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Sutherland-Hodgeman Polygon Clipping
Top
Bottom
Right
Left