CPSC 441 Computer Graphics: Clipping Lines Jinxiang Chai

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CPSC 441 Computer GraphicsClipping
LinesJinxiang Chai
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Why Clip?

• We do not want to waste time drawing objects that
  are outside of viewing window (or clipping window)

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Clipping Points

• Given a point (x, y) and clipping window (xmin,
  ymin), (xmax, ymax), determine if the point
  should be drawn

(xmax,ymax)
(x,y)
(xmin,ymin)
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Clipping Points

• Given a point (x, y) and clipping window (xmin,
  ymin), (xmax, ymax), determine if the point
  should be drawn

(xmax,ymax)
(x,y)
xminltxltxmax?
(xmin,ymin)
yminltyltymax?
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Clipping Points

• Given a point (x, y) and clipping window (xmin,
  ymin), (xmax, ymax), determine if the point
  should be drawn

(xmax,ymax)
(x2,y2)
(x1,y1)
xminltxltxmax?
(xmin,ymin)
yminltyltymax?
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Clipping Points

• Given a point (x, y) and clipping window (xmin,
  ymin), (xmax, ymax), determine if the point
  should be drawn

(xmax,ymax)
(x2,y2)
(x1,y1)
xminltx1ltxmax Yes
(xmin,ymin)
yminlty1ltymax Yes
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Clipping Points

• Given a point (x, y) and clipping window (xmin,
  ymin), (xmax, ymax), determine if the point
  should be drawn

(xmax,ymax)
(x2,y2)
(x1,y1)
xminltx2ltxmax No
(xmin,ymin)
yminlty2ltymax No
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Clipping Lines
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Clipping Lines
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Clipping Lines

• Given a line with end-points (x0, y0), (x1, y1)
• and clipping window (xmin, ymin), (xmax, ymax),
• determine if line should be drawn and clipped
  end-points of line to draw.

(xmax,ymax)
(x1, y1)
(x0, y0)
(xmin,ymin)
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Clipping Lines
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Outline

• Simple line clipping algorithm
• Cohen-Sutherland
• Liang-Barsky

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Clipping Lines Simple Algorithm

• If both end-points inside rectangle, draw line
• If one end-point outside,
• intersect line with all edges of rectangle
• clip that point and repeat test

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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Intersecting Two Lines
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Intersecting Two Lines
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Intersecting Two Lines
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Intersecting Two Lines
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Intersecting Two Lines
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Intersecting Two Lines
Substitute t or s back into equation to find
intersection
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm
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Clipping Lines Simple Algorithm

• Lots of intersection tests makes algorithm
  expensive
• Complicated tests to determine if intersecting
  rectangle
• Is there a better way?

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Trivial Accepts

• Big Optimization trivial accepts/rejects
• How can we quickly decide whether line segment is
  entirely inside window
• Answer test both endpoints

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Trivial Accepts

• Big Optimization trivial accepts/rejects
• How can we quickly decide whether line segment is
  entirely inside window
• Answer test both endpoints

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Trivial Rejects

• How can we know a line is outside of the window
• Answer both endpoints on wrong side of same
  edge, can trivially reject the line

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Trivial Rejects

• How can we know a line is outside of the window
• Answer both endpoints on wrong side of same
  edge, can trivially reject the line

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Cohen-Sutherland Algorithm

• Classify p0, p1 using region codes c0, c1
• If , trivially reject
• If , trivially accept
• Otherwise reduce to trivial cases by splitting
  into two segments

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Cohen-Sutherland Algorithm

• Every end point is assigned to a four-digit
  binary value, i.e. Region code
• Each bit position indicates whether the point
  is inside or outside of a specific window edges

bit 4
bit 3
bit 2
bit 1
left
right
bottom
top
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Cohen-Sutherland Algorithm
bit 4
bit 3
bit 2
bit 1
left
right
bottom
top
top
right
left
Region code?
bottom
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Cohen-Sutherland Algorithm
bit 4
bit 3
bit 2
bit 1
left
right
bottom
top
top
right
left
0010
bottom
?
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Cohen-Sutherland Algorithm
bit 4
bit 3
bit 2
bit 1
left
right
bottom
top
top
right
left
0010
bottom
0100
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm

• Classify p0, p1 using region codes c0, c1
• If , trivially reject
• If , trivially accept
• Otherwise reduce to trivial cases by splitting
  into two segments

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Cohen-Sutherland Algorithm

• Classify p0, p1 using region codes c0, c1
• If , trivially reject
• If , trivially accept
• Otherwise reduce to trivial cases by splitting
  into two segments

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Cohen-Sutherland Algorithm

• Classify p0, p1 using region codes c0, c1
• If , trivially reject
• If , trivially accept
• Otherwise reduce to trivial cases by splitting
  into two segments

Line is outside the window! reject
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Cohen-Sutherland Algorithm

• Classify p0, p1 using region codes c0, c1
• If , trivially reject
• If , trivially accept
• Otherwise reduce to trivial cases by splitting
  into two segments

Line is inside the window! draw
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Cohen-Sutherland Algorithm

• Classify p0, p1 using region codes c0, c1
• If , trivially reject
• If , trivially accept
• Otherwise reduce to trivial cases by splitting
  into two segments

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Window Intersection

• (x1, y1), (x2, y2) intersect with vertical edge
  at xright
• yintersect y1 m(xright x1)
• where m(y2-y1)/(x2-x1)
• (x1, y1), (x2, y2) intersect with horizontal edge
  at ybottom
• xintersect x1 (ybottom y1)/m
• where m(y2-y1)/(x2-x1)

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Example 1
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Example 1
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Example 1
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Example 1
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Example 2
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Example 2
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Example 2
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Example 2
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Example 2
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Example 2
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Example 2
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Example 3
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Example 3
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Example 3
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Example 3
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Cohen-Sutherland Algorithm

• Extends easily to 3D line clipping
• 27 regions
• 6 bits

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Cohen-Sutherland Algorithm

• Use region codes to quickly eliminate/include
  lines
• Best algorithm when trivial accepts/rejects are
  common
• Must compute viewing window clipping of remaining
  lines
• Non-trivial clipping cost
• Redundant clipping of some lines
• More efficient algorithms exist

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Liang-Barsky Algorithm

• Parametric definition of a line
• x x1 u?x
• y y1 u?y
• ?x (x2-x1), ?y (y2-y1), 0ltult1
• Lines are oriented classify lines as moving
  inside to out or outside to in
• Goal find range of u for which x and y both
  inside the viewing window

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Liang-Barsky Algorithm

• For lines starting outside of boundary, update
  its starting point (u1)
• For lines starting inside of boundary, update end
  point (u2)
• For lines paralleling the boundaries and outside
  window, reject it.

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Liang-Barsky Algorithm

• For lines starting outside of boundary, update
  its starting point (u1)
• For lines starting inside of boundary, update end
  point (u2)
• For lines paralleling the boundaries and outside
  window, reject it.

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Liang-Barsky Algorithm

• Mathematically
• xmin lt x1 u?x lt xmax
• ymin lt y1 u?y lt ymax
• Rearranged
• 1 u(-?x) lt (x1 xmin)
• 2 u(?x) lt (xmax x1)
• 3 u(-?y) lt (y1 ymin)
• 4 u(?y) lt (ymax y1)
• gen u(pk) lt (qk), k1,2,3,4

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Liang-Barsky Algorithm

• Rules
• pk 0 the line is parallel to boundaries
• If for that same k, qk lt 0, its outside
• Otherwise its inside
• pk lt 0 the line starts outside this boundary
• rk qk/pk
• u1 max(0, rk, u1) /update starting point/
• pk gt 0 the line starts inside the boundary
• rk qk/pk
• u2 min(1, rk, u2) / update end point/
• If u1 gt u2, the line is completely outside

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Liang-Barsky Algorithm

• Current clipped line (u10,u21)

(2,1)
(-.5,-.5)
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Liang-Barsky Algorithm

• Current clipped line (u10,u21)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p1 -?x -2.5lt0
(-.5,-.5)
q1 (x1-xmin)-.5
r1 q1/p10.2
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Liang-Barsky Algorithm

• Current clipped line (u10,u21)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p1 -?x -2.5lt0
(-.5,-.5)
q1 (x1-xmin)-.5
line starts outside this boundary
r1 q1/p10.2
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Liang-Barsky Algorithm

• Current clipped line (u10,u21)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p1 -?x -2.5lt0
(-.5,-.5)
q1 (x1-xmin)-.5
r1 q1/p10.2
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Liang-Barsky Algorithm

• Current clipped line (u10,u21)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p1 -?x -2.5lt0
(-.5,-.5)
q1 (x1-xmin)-.5
r1 q1/p10.2
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Liang-Barsky Algorithm

• Current clipped line (u10,u21)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p1 -?x -2.5lt0
(-.5,-.5)
q1 (x1-xmin)-.5
r1 q1/p10.2
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Liang-Barsky Algorithm

• Current clipped line (u10,u21)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p1 -?x -2.5lt0
(-.5,-.5)
q1 (x1-xmin)-.5
r1 q1/p10.2
u1 max(0, r1, u1)
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Liang-Barsky Algorithm

• Current clipped line (u10.2,u21)

(2,1)

• u1ltu2?
• yes continue
• no the line is completely outside window

(-.5,-.5)
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Liang-Barsky Algorithm

• Current clipped line (u10.2,u21)

(2,1)
(-.5,-.5)
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Liang-Barsky Algorithm

• Current clipped line (u10.2,u21)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p2 ?x 2.5gt0
(-.5,-.5)
q2 (xmax-x1)1.5
r2 q2/p20.6
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Liang-Barsky Algorithm

• Current clipped line (u10.2,u21)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p2 ?x 2.5gt0
(-.5,-.5)
q2 (xmax-x1)1.5
r2 q2/p20.6
u2 min(1, r2, u2)
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Liang-Barsky Algorithm

• Current clipped line (u10.2,u20.6)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p2 ?x 2.5gt0
(-.5,-.5)
q2 (xmax-x1)1.5
r2 q2/p20.6
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Liang-Barsky Algorithm

• Current clipped line (u10.2,u20.6)

(2,1)

• u1ltu2?
• yes continue
• no the line is completely outside window

(-.5,-.5)
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Liang-Barsky Algorithm

• Current clipped line (u10.2,u20.6)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p3 -?y -1.5lt0
(-.5,-.5)
q3 (y1-ymin)0.5
r3 q3/p30.333
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Liang-Barsky Algorithm

• Current clipped line (u10.2,u20.6)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p3 -?y -1.5lt0
(-.5,-.5)
q3 (y1-ymin)0.5
r3 q3/p30.333
u1 max(0, r3, u1)
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Liang-Barsky Algorithm

• Current clipped line (u10.333,u20.6)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p3 -?y -1.5lt0
(-.5,-.5)
q3 (y1-ymin)0.5
r3 q3/p30.333
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Liang-Barsky Algorithm

• Current clipped line (u10.333,u20.6)

(2,1)
(-.5,-.5)
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Liang-Barsky Algorithm

• Current clipped line (u10.333,u20.6)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p3 -?y -1.5lt0
(-.5,-.5)
q3 (y1-ymin)0.5
r3 q3/p30.333
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Liang-Barsky Algorithm

• Current clipped line (u10.333,u20.6)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p4 ?y 1.5gt0
(-.5,-.5)
q4 (ymax-y1)1.5
r4 q4/p41
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Liang-Barsky Algorithm

• Current clipped line (u10.333,u20.6)

(2,1)
?x (x2-x1)2.5
?y (y2-y1)1.5
p4 ?y 1.5gt0
(-.5,-.5)
q4 (ymax-y1)1.5
r4 q4/p41
u2 min(1, r4, u2)
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Liang-Barsky Algorithm

• Check the left edge (u10.333,u20.6)

(2,1)
u1ltu2
(-.5,-.5)
xnew1x1?xu1
xnew2x2?xu2
ynew1y1?yu1
ynew2y2?yu2
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
u1gtu2!!
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm
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Liang-Barsky Algorithm

• Faster than Cohen-Sutherland
• Extension to 3D is easy
• - Parametric representation for 3D lines
• - Compute u1,u2 based on the intersection
  between line and plane

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Comparison

• Cohen-Sutherland
• Repeated clipping is expensive
• Best used when trivial acceptance and rejection
  is possible for most lines
• Liang-Barsky
• Computation of t-intersections is cheap (only one
  division)
• Computation of (x,y) clip points is only done
  once
• Algorithm doesnt consider trivial
  accepts/rejects
• Best when many lines must be clipped
• Nicholl et al. Fastest, but doesnt do 3D

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Curve Clipping
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Curve Clipping
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Curve Clipping

• Approximate a curve using a set of straight-line
  segments
• Apply line clipping for curve clipping

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Next Lecture

• Polygon fill-area clipping

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Next Lecture

• Polygon fill-area clipping