Clipping

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Clipping
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Transformation Sequence
Normalized Device Coords.
Screen Coords.
Clip Coords.
Object Coords.
Eye Coords.
Implementation 4 x 4 matrix multiplication in
homogeneous coords.
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Culling and Clipping

• What cant we see?
• anything occluded by another object (HSR)
• anything outside view volume
• Today clipping

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Clipping Against a Rectangular RegionMultiple
Cases
B
F
E
C
G
A
D
H
Clip Rectangle
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Division of Space
Clip Region
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Cohen-Sutherland ClippingOutcodes
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Cohen-Sutherland ClippingRegion Outcodes
1001
1000
1010
0001
0010
0000
0100
0110
0101
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Cohen-Sutherland ClippingTrivial Acceptance
O(P0) O(P1) 0
1001
1000
1010
P1
0001
0010
0000
P0
0100
0110
0101
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Cohen-Sutherland ClippingTrivial Rejection
O(P0) O(P1) ? 0
P0
1001
1000
1010
P1
P0
0001
0010
0000
P0
P1
P1
0100
0110
0101
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Cohen-Sutherland Clipping O(P0) 0 , O(P1) ? 0
1001
1000
1010
P0
0001
0010
0000
P1
P1
P0
P0
0100
0110
0101
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 ClippingExample 1
A
1001
1000
1010
B
C
0001
0010
0000
0100
0110
0101
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Cohen-Sutherland ClippingExample 1
1001
1000
1010
B
C
0001
0010
0000
0100
0110
0101
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Chopping at each boundary
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Cohen-Sutherland ClippingExample 2
1001
1000
1010
E
D
0001
0010
0000
C
B
A
0100
0110
0101
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Cohen-Sutherland ClippingExample 2
1001
1000
1010
E
D
0001
0010
0000
C
B
0100
0110
0101
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Cohen-Sutherland ClippingExample 2
1001
1000
1010
D
0001
0010
0000
C
B
0100
0110
0101
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Cohen-Sutherland ClippingExample 2
1001
1000
1010
0001
0010
0000
C
B
0100
0110
0101
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Cohen-Sutherland ClippingAdvantages/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

• CyrusBeck, LiangBarsky
• 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 CohenSutherland
• try to increase number of trivial rejection cases

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Parametric Representation of Lines
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Liang-Barsky Clipping

• Consider the parametric form of a line segment
• We can distinguish between the cases by looking
  at the ordering of the values of a where the line
  determined by the line segment crosses the lines
  that determine the window

p(a) (1-a)p1 ap2 1 ? a ? 0
p2
p1
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Liang-Barsky Parametric Clipping
Edge Ei
Inside
Outside
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Potentially Entering (PE) and Potentially Leaving
(PL) Points
Edge Ei
Edge Ei
Inside
Outside
Inside
Outside
Potentially Entering (PE) Point
Potentially Leaving (PL) Point
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Liang-Barsky ClippingComputing the Intersection
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Liang-Barsky ClippingPotentially Leaving vs.
Potentially Entering
PE
PL
PL
PL
PE
PL
PE
PE
Inside
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Liang-Barsky ClippingAlgorithm Strategy

• Find the largest PE greater than zero.
• Find the smallest PL less than one.
• Reject the segment if PE gt PL.

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Liang-Barsky ClippingPseudocode of Algorithm
for (each line segment to be clipped) alpha_E0
alpha_L1 for (each candidate intersection with
a clip edge) if (NiD!0) /edges not
parallel to line/ calculate alpha use
sign of NiD 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
gt alpha_L) return NULL else return
P(alpha_E) and P(alpha_L) as clip intersections
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Polygon ClippingConvex Polygons
5
4
4
5
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 ClippingThe Convexity Problem
Single object becomes multiple objects.
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Sutherland-Hodgeman Polygon Clipping