Dr' Scott Schaefer

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Clipping Lines

• Dr. Scott Schaefer

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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 Yes
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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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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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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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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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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
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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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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Cohen-Sutherland Algorithm
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Liang-Barsky Algorithm

• Uses parametric form of line for clipping
• Lines are oriented
• Classify lines as moving inside to out or
  outside to in
• Dont find actual intersection points
• Find parameter values on line to draw

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Intersecting Two Parametric Lines
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Intersecting Two Parametric Lines
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Intersecting Two Parametric Lines
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Intersecting Two Parametric Lines
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Intersecting Two Parametric Lines
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Intersecting Two Parametric Lines
Substitute t or s back into equation to find
intersection
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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
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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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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

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Line Clipping Considerations

• Just clipping end-points does not produce the
  correct results inside the window
• Must also update sum in midpoint algorithm
• Clipping against non-rectangular polygons is also
  possible but seldom used