Overview
Lecture 5
Clipping
ƒ Cohen-Sutherland line clipping algorithm
ƒ Liang-Barsky line clipping algorithm
ƒ Sutherland-Hogeman polygon clipping
Line and Polygon Clipping
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Clipping Algorithms
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Cohen-Sutherland Line-Clipping
Line Clipping:
1.
1000
1010
0001
0000
0010
If C0 ∧ Cend ≠ 0 then
P0Pend is trivially rejected
0101
0100
0110
3.
If C0 ∨ Cend = 0 then
P0Pend is trivially accepted
C0 =
4.
Otherwise subdivide and
go to step 1 with new segment.
- Oldest and most commonly used
ƒ Nicholl-Lee-Nicholl (encoding) (more efficient)
ƒ Liang-Barsky (parametric)
2.
- More efficient than Cohen-Sutherland
Polygon Clipping:
ƒ Sutherland-Hodgeman (divide and conquer
strategy)
ƒ Weiler-Atherton (modified for concave polygons)
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Pedher Johansson
Department of Computing Science, Umeå University
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Cohen-Sutherland Line-Clipping
1)
2)
3)
4)
5
A3D3
A3C3
A3B3
accept
1)
2)
3)
4)
5)
A2E2
B2E2
1001
B2D2
B2C2
accept A1
0001
C1
1010
D2
C2
0010
B2 0000
A3
A2 0101
C3
B3
0100
0110
D3
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Pedher Johansson
Department of Computing Science, Umeå University
Cend = Bit code of Pend
Pedher Johansson
Department of Computing Science, Umeå University
ƒ Will do unnecessary clipping.
ƒ Not the most efficient.
ƒ Clipping and testing are done in fixed
order.
ƒ Efficient when most of the lines to be
clipped are either rejected or accepted
(not so many subdivisions).
ƒ Easy to program.
ƒ Parametric clipping are more efficient.
E2
B1 1000
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Bit code of P0
Cohen-Sutherland Line-Clipping
Clip order: Left, Right, Bottom, Top
1) A1C1
2) B1C1
3) reject
Encode end points
Bit 0 = point is left of window
Bit 1 = point is right of window
Bit 2 = point is below window
Bit 3 = point is above window
1001
ƒ Cohen-Suterland (encoding)
3
Pedher Johansson
Department of Computing Science, Umeå University
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Pedher Johansson
Department of Computing Science, Umeå University
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Parametric form
Liang-Barsky Line-Clipping
ƒ A line segment with endpoints
ƒ More efficient than Cohen-Sutherland
(x0, y0) and (xend, yend)
ƒ A line is inside the clipping region for
values of u such that:
Δ x = xend – x0
xwmin ≤ x0 + uΔx ≤ xwmax
ywmin ≤ y0 + uΔy ≤ ywmax
Δ y = yend – y0
we can describe in the parametric form
x = x0 + uΔx
y = x0 + uΔy
0≤u≤1
where
ƒ Can be described as
u pk ≤ qk,
k = 1, 2, 3, 4
Δx = xend – x0
Δy = yend – y0
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Pedher Johansson
Department of Computing Science, Umeå University
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Liang-Barsky Line-Clipping
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Pedher Johansson
Department of Computing Science, Umeå University
Liang-Barsky Line-Clipping
The infinitely line intersects the clip
region edges when:
ƒ When pk < 0, as u increases
- line goes from outside to inside - entering
ƒ When pk > 0,
p1 = −Δx q1 = x0 − xwmin
p2 = Δx q2 = xwmax − x0
q
uk = k where
p3 = −Δy q3 = y0 − ywmin
pk
p4 = Δy q4 = ywmax − y0
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- line goes from inside to outside - exiting
Left boundary
Right boundary
Bottom boundary
Top boundary
Pedher Johansson
Department of Computing Science, Umeå University
ƒ When pk = 0,
- line is parallel to an edge
ƒ If there is a segment of the line inside the
clip region, a sequence of infinite line
intersections must go: entering, entering,
exiting, exiting
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Liang-Barsky Line-Clipping
Enter
Exit
Enter
11
1. Set umin = 0 and umax = 1.
2. Calculate the u values:
3. If u < umin or u > umax ignore it.
Otherwise classify the u values as
entering or exiting.
4. If umin < umax then draw a line from:
Exit
Clip region
Enter
( x0 + Δx · umin, y0 + Δy · umin ) to
( x0 + Δx · umax, y0 + Δy · umax )
Enter
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Pedher Johansson
Department of Computing Science, Umeå University
Liang-Barsky Line-Clipping
Exit
Exit
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Pedher Johansson
Department of Computing Science, Umeå University
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Example
Liang-Barsky
10,10
0,10
Liang-Barsky Line-Clipping
Pend(15,9)
P0(-5,3)
ƒ We have umin = 1/4 and umax = 3/4
0,0
uleft =
uright
10,0
q1 x0 − xwmin
1
−5−0
=
=
=
p1
− (15 − (−5)) 4
− Δx
xw − x 10 − (−5) 3
q
= 2 = max 0 =
=
p2
15 − (−5) 4
Δx
Exiting
u < 0 then ignore
q4 ywmax − y0 10 − 3 7
=
=
=
9−3 6
Δy
p4
u > 1 then ignore
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Computer Graphics & Visualization
Example
Liang-Barsky
0,10
Δx
Department of Computing Science, Umeå University
Pend(2,14)
- compute endpoints by substituting u values
ƒ Draw a line from
(-5+(20)·(1/4), 3+(6)·(1/4))
to
(-5+(20)·(3/4), 3+(6)·(3/4))
14
10,10
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ƒ We have umin = 4/5 and umax = 2/3
0,0
q1 x0 − xwmin
4
−8−0
=
=
=
p1
− Δx
− (2 − (−8)) 5
uright =
q2 xwmax − x0 10 − (−8) 9
=
=
=
p2
2 − (−8) 5
Δx
ubottom =
utop =
15
10,0
Entering
Pend - P0 = (2+8, 14-2) = (10, 12)
umin = 4/5
ƒ umin > umax ,
there is no line segment do draw
u > 1 then ignore
q3 y0 − ywmin
2−0
1
=
=
= − u < 0 then ignore
6
p3
− Δy
− (14 − 2)
q4 ywmax − y0 10 − 2 2
=
=
=
14 − 2 3
Δy
p4
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Exiting
umax = 2/3
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Department of Computing Science, Umeå University
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Nicholl-Lee-Nicholl Line Clipping
L
B
If P0 is to the left:
P0
edge region
If P0 is to the left and above:
P0
L
corner region
T
L
P0
ƒ Find position of Pend relative to P0.
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R
P0
LT
inside
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Department of Computing Science, Umeå University
T
If P0 inside and Pend outside:
P0
P0
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Nicholl-Lee-Nicholl Line Clipping
ƒ Avoids multiple clipping of an individual
line by creating more regions.
ƒ Only three regions need to be considered.
17
Pedher Johansson
Department of Computing Science, Umeå University
Liang-Barsky Line-Clipping
P0(-8,2)
uleft =
Δy
ƒ If umin < umax , there is a line segment
umax = 3/4
q
y − ywmin
3−0
1
ubottom = 3 = 0
=
=−
2
p3
− Δy
− (9 − 3)
utop =
Pend - P0 = (15+5,9-3) = (20,6)
umin = 1/4
Entering
LR
L
L
LB
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Department of Computing Science, Umeå University
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TR
T
LB
TB
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Department of Computing Science, Umeå University
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Nicholl-Lee-Nicholl Line Clipping
Suterland-Hodgeman
Polygon Clipping
To determine region
Four test cases:
1.
2.
3.
4.
ƒ compare slopes, and
ƒ boundaries of the NLN region.
First vertex inside and the second outside (in-out pair)
Both vertices inside clip window
First vertex outside and the second inside (out-in pair)
Both vertices outside the clip window
ƒ Concave polygons may be displayed with extra lines.
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Pedher Johansson
Department of Computing Science, Umeå University
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Pedher Johansson
Department of Computing Science, Umeå University
Weiler-Atherton
Polygon Clipping
ƒ Clips concave polygons correctly.
ƒ Instead of always going around the
polygon edges, we also, want to follow
window boundaries.
ƒ For an outside-to-inside pair of vertices,
follow the polygon boundary.
ƒ For an inside-to-outside pair of vertices,
follow the window boundary in a clockwise
direction.
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Pedher Johansson
Department of Computing Science, Umeå University
4