Two Dimensional Viewing
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Graphics Pipeline
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Graphics Pipeline
Window-to-Viewport Mapping
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Viewing Transformation
Clipping
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Clipping Points
Clipping Lines
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Clipping Polygons
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Cohen-Sutherland Algorithm
Liang-Barsky Algorithm
Sutherland-Hodgeman Algorithm
Weiler-Atherton Algorithm
Text Clipping
Computer Graphics – Lecture 5
Computer Graphics – Lecture 5
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Position of a Viewport within a Display
Window
Window-to-Viewport Mapping (1)

We need to map from the clipping window to
viewport

Clipping window specified in the World
Coordinate System
Viewport specified in the Device (Screen)
Coordinate System
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Computer Graphics – Lecture 5
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Computer Graphics – Lecture 5
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1
Window-to-Viewport Mapping (2)
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Clipping
 Clipping Rectangle
defined by xl, xr, yt, yb
Mapping done by using geometric transformations:
1. Translation: bring bottom-left corner of clipping
window to origin
=> Multiply by Translation Matrix T(–xc, –yc)
2. Scaling: Resize clipping window to size of viewport
=> Multiply by Scaling Matrix S(wv / wc, hv / hc)
3. Translation: bring bottom-left corner of clipping
window to bottom-left corner of viewport
=> Multiply by Translation Matrix T(xv, yv)
 Clipping Points: easy!
 Clipping Lines: more
difficult
Composite Transformation Matrix:
M = T(xv, yv) · S(wv / wc, yv / yc) · T(–xc, –yc)
Computer Graphics – Lecture 5
Cohen-Sutherland Line Clipping
Algorithm (1)
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Computer Graphics – Lecture 5
Cohen-Sutherland Line Clipping
Algorithm (2)
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 Assign a 4 bit region code to each region: b4b3b2b1
b1 = 1, if region is to the left
of the left boundary
b2 = 1, if region is to the right
of the right boundary
b3 = 1, if region is below
bottom boundary
b4 = 1, if region is above
top boundary
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Computer Graphics – Lecture 5
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2
Cohen-Sutherland Line Clipping
Algorithm (3)
A.
B.
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C.
Cohen-Sutherland Line Clipping
Algorithm (5)
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The worst case requires four clips
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Cohen-Sunderland is not
the most efficient
algorithm as it can end
up doing needless
clippings
Still widely used since
widely known
Extension to 3D
straightforward

 Find Region Codes (C1 and C2) G
for line
 Remove line from input list
 If C1 OR C2 = 0000, then add line
to output list
 Else If C1 AND C2 = 0000, then
How to trivially reject line below, to right, and to left:
C1 AND C2 =
C1 AND C2 =
C1 AND C2 =
How to generalize these four cases?
C1 AND C2 is not equal to 0000
The rest are difficult:
C1 AND C2 = 0000
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Cohen-Sutherland Line Clipping
Algorithm (4)
 Start with an input list of lines
(endpoints)
 While input list is not empty
If C1 OR C2 = 0000, then
trivially accept
How to trivially reject a line
that has both points above
the top?
C1 AND C2 = 1xxx
Computer Graphics – Lecture 5
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E
F
I
H
J
 Find intersection of line with an
edge (top, bottom, left, right order)
 Add intersection point and interior
point to input list
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End
Computer Graphics – Lecture 5
Cyrus-Beck and Liang-Barsky
Parametric Line Clipping Algorithms
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Cyrus-Beck Algorithm
A
More efficient
Can clip against convex polygon clip region
Liang-Barsky
Like Cyrus-Beck, but faster for rectangular
clip regions
B
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Computer Graphics – Lecture 5
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Liang-Barsky Line Clipping
Algorithm (1)
Liang-Barsky Line Clipping
Algorithm (2)
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xl ≤ x1 + uΔx ≤ xr
yb ≤ y1 + uΔy ≤ yt
 Based on the parametric representation of a line
Rewrite as four inequalities:
(x1,y1)
upk ≤ qk, where k = 1, 2, 3, 4
p1 = –Δx,
p2 = Δx,
p3 = –Δy,
p4 = Δy,
(x2,y2)
Computer Graphics – Lecture 5
Liang-Barsky Line Clipping
Algorithm (3)
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Computer Graphics – Lecture 5
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Each value of k corresponds to one
boundary:
 k = 1 corresponds to left boundary
 k = 2 corresponds to right boundary
 k = 3 corresponds to bottom
boundary
 k = 4 corresponds to top boundary
If qk < 0, then p1 is outside kth
boundary
If qk ≥ 0, then p1 is inside or on kth
boundary
If pk < 0, then line goes in the direction
from outside to inside the boundary
If pk > 0, then line goes in the direction
from inside to outside the boundary
Computer Graphics – Lecture 5
q 1 = x 1 – xl
q 2 = x r – x1
q3 = y1 – yb
q4 = yt – y1
Liang-Barsky Line Clipping
Algorithm (4)
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If pk ≠ 0, then the
intersection of the line
with the kth boundary is
at
rk = qk / pk
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For each line we want
to find u1 and u2 that lie
in the clip region
Computer Graphics – Lecture 5
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Liang-Barsky Line Clipping Algorithm (5)
For each line segment:
 u1 = 0; u2 = 1 (start with original endpoints)
 k = 1
 While still need to clip and k ≤ 4
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Compute pk and qk
If pk = 0 and qk < 0, then reject line and stop clipping
Else
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If u1 > u2 then reject line and stop clipping
k = k + 1;
End
If line not rejected, u1 and u2 are endpoints of
clipped line
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Liang-Barsky computes rk = qk / pk for a line
needing clipping, which is faster than computing
the intersection of lines in Cohen-Sutherland
Liang-Barsky does not have a trivial accept or
reject
If most lines can be trivially accepted or rejected
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Example Run:
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Liang-Barsky versus Cohen-Sutherland
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u1 = 0, u2 = 1, p1 < 0, q1 < 0
u1 = r1, u2 = 1, p2 > 0, q2 > 0
u1 = r1, u2 = 1, p3 > 0, q3 > 0
u1 = r1, u2 = r3, p4 < 0, q4 > 0
u1 = r1, u2 = r3
rk = qk / pk
If pk < 0, u1 = max{u1, rk}
Else u2 = min{u2, rk}
Computer Graphics – Lecture 5
Liang-Barsky Line Clipping
Algorithm (6)
Use Cohen-Sutherland
Line Clipping Using Nonrectangular
Polygon Clip Windows
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Cyrus-Beck and Liang-Barsky can be
extended to clip lines against convex
polygon windows
For concave-polygon clipping regions,
add dummy edges to modify clipping
regions to convex-polygon shape
Else

Use Liang-Barsky
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Computer Graphics – Lecture 5
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Clipping Polygons
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Want to clip polygons
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Sutherland-Hodgeman Polygon
Clipper Algorithm (1)
Divide and Conquer algorithm
Clipping against clip region implemented by clip
against each region edge
Suitable for parallel and pipeline implementation
If just clip lines
top
Computer Graphics – Lecture 5
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Sutherland-Hodgeman Polygon
Clipper Algorithm (2)
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Consider a polygon described
v3
by a sequence of vertices
V = (v1, v2, v3, v4, …, vn)
Want to clip against clip region
v7
to get clipped polygon
v2
C = (c1, c2, c3, …, cm)
Let E be the edge list of V
E = {sp | s = vi, p = vi+1} U {vnv1}
Want to get updated edge list:
K = {sp | s = ci, p = ci+1} U {cnc1}
Computer Graphics – Lecture 5
Computer Graphics – Lecture 5
Sutherland-Hodgeman Polygon
Clipper Algorithm (3)
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v4
v6
v5
v1
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Sutherland-Hodgeman Polygon
Clipper Algorithm – Example
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Weiler-Atherton Polygon Clipping
Algorithm (1)
V = (a, b, c, d, e, f, g)
E = (ab, bc, cd, de, ef, fg, ga)

Updated E after clipping by:
 Right boundary
E=
 Bottom boundary
E=
 Left boundary
E=
 Top boundary
E=
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Computer Graphics – Lecture 5
Weiler-Atherton Polygon Clipping
Algorithm (2)
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Computer Graphics – Lecture 5
Weiler-Atherton Polygon Clipping
Algorithm – Pseudocode
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Begin
position = start
save = off
trace-subject(position)
End
Trace-subject(position)
If hit clip
if (entering clip and subject not saved)
save = on
trace-subject(current)
else if (leaving clip & clip right not
Trace-clip(position)
saved)
If hit subject
save = on
if subject right not saved
push current onto stack
turn on save
trace-clip(current)
trace-subject(current)
else
else
save = off
turn off save
if stack empty then stop
output region just completed
else popstack(current)
if stack empty then stop
trace-subject(current)
else popstack(current)
end
trace-subject(current)
end
While traversing the subject polygon:
 When going from outside the clip polygon to inside,
follow the subject polygon boundary in clockwise
direction
 When going from inside the clip polygon to outside,
follow the clip polygon boundary in clockwise direction
Computer Graphics – Lecture 5
Very general
Can clip convex polygon by convex polygon
Can clip concave polygon by a convex polygon
Clip region is called clip polygon
Polygon to be clipped is called subject polygon
Traces clockwise around the polygons
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Weiler-Atherton Polygon Clipping
Algorithm – Example (1)
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Weiler-Atherton Polygon Clipping
Algorithm – Example (2)
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Counterclockwise
processing
Computer Graphics – Lecture 5
Text Clipping
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Computer Graphics – Lecture 5
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All-or-none string-clipping strategy
All-or-none character-clipping strategy
Clip the components of individual
characters
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