Clipping

1
Clipping

• CS418 Computer Graphics
• John C. Hart

2
Graphics Pipeline
ModelCoords
ModelXform
WorldCoords
ViewingXform
ViewingCoords
StillClipCoords.
Clipping
ClipCoords.
Homogeneous Divide
Window Coordinates
Window to Viewport
Viewport Coordinates
3
Why Clip?

• Why not just transform all triangles to the
  screen and just ignore pixels off the screen?
• Takes time to rasterize a triangle
• Very small number of triangles fall within the
  viewing frustum
• Output may not go directly to screen

4
Outcodes
1000
1001
1010
y top

• Cohen-Sutherland
• Assign segment endpoints a bitcode
• b3b2b1b0
• b0 x lt left
• b1 x gt right
• b2 y lt bottom
• b3 y gt top
• Let o0 outcode(x0,y0), o1 outcode(x1,y1)
• o0 o1 0 segment visible
• o0 0, o1 ? 0 segment must be clipped

0001
0010
0000
y bottom
0100
0101
0110
x left
x right
5
Outcodes
1000
1001
1010
y top

• Cohen-Sutherland
• Assign segment endpoints a bitcode
• b3b2b1b0
• b0 x lt left
• b1 x gt right
• b2 y lt bottom
• b3 y gt top
• Let o0 outcode(x0,y0), o1 outcode(x1,y1)
• o0 o1 0 segment visible
• o0 0, o1 ? 0 segment must be clipped

0001
0010
0000
y bottom
0100
0101
0110
x left
x right
6
Outcodes
1000
1001
1010
y top

• Cohen-Sutherland
• Assign segment endpoints a bitcode
• b3b2b1b0
• b0 x lt left
• b1 x gt right
• b2 y lt bottom
• b3 y gt top
• Let o0 outcode(x0,y0), o1 outcode(x1,y1)
• o0 o1 0 segment visible
• o0 0, o1 ? 0 segment must be clipped
• o0 o1 ? 0 segment can be ignored

0001
0010
0000
y bottom
0100
0101
0110
x left
x right
7
Outcodes
1000
1001
1010
y top

• Cohen-Sutherland
• Assign segment endpoints a bitcode
• b3b2b1b0
• b0 x lt left
• b1 x gt right
• b2 y lt bottom
• b3 y gt top
• Let o0 outcode(x0,y0), o1 outcode(x1,y1)
• o0 o1 0 segment visible
• o0 0, o1 ? 0 segment must be clipped
• o0 o1 ? 0 segment can be ignored
• o0 o1 0 segment might need clipping

0001
0010
0000
y bottom
0100
0101
0110
x left
x right
8
Outcodes
1000
1001
1010
y top

• Cohen-Sutherland
• Assign segment endpoints a bitcode
• b3b2b1b0
• b0 x lt left
• b1 x gt right
• b2 y lt bottom
• b3 y gt top
• Let o0 outcode(x0,y0), o1 outcode(x1,y1)
• o0 o1 0 segment visible
• o0 0, o1 ? 0 segment must be clipped
• o0 o1 ? 0 segment can be ignored
• o0 o1 0 segment might need clipping

0001
0010
0000
y bottom
0100
0101
0110
x left
x right
9
Intersecting Lines
tgt1
t1
(x1,y1)
t0
(x(t),y(t))
tlt0

• Parametric representation of a line segment
• x(t) x0 t (x1 x0)
• y(t) y0 t (y1 y0)

(x0,y0)
10
Intersecting Lines
(x1,y1)
y top

• Parametric representation of a line segment
• x(t) x0 t (x1 x0)
• y(t) y0 t (y1 y0)
• Plug in clipping window edge to find t
• top y0 t (y1 y0)
• t (top y0)/(y1 y0)

(x0,y0)
11
Serial Clipping
1000
1001
1010
y top
0001
0010
y bottom
0100
0101
0110
x left
x right
12
Serial Clipping
1000
1001
1010
y top

• First clip 0001
• Move (x0,y0) to (left,)

0001
0010
y bottom
0100
0101
0110
x left
x right
13
Serial Clipping
1000
1001
1010
y top

• First clip 0001
• Move (x0,y0) to (left,)
• Then clip 0010
• Move (x1,y1) to (right,)

0001
0010
y bottom
0100
0101
0110
x left
x right
14
Serial Clipping
1000
1001
1010
y top

• First clip 0001
• Move (x0,y0) to (left,)
• Then clip 0010
• Move (x1,y1) to (right,)
• Then clip 0100
• Move (x0,y0) again, now to (,bottom)

0001
0010
y bottom
0100
0101
0110
x left
x right
15
Serial Clipping
1000
1001
1010
y top

• First clip 0001
• Move (x0,y0) to (left,)
• Then clip 0010
• Move (x1,y1) to (right,)
• Then clip 0100
• Move (x0,y0) again, now to (,bottom)
• Finally clip 1000
• Move (x1,y1) again, now to (,top)

0001
0010
y bottom
0100
0101
0110
x left
x right
16
Parametric Clipping
tright
ttop
y top

• Cohen-Sutherland iterates through each edge
  intersection and computes them even if they are
  later clipped
• Liang-Barsky computes all at once postponing
  division to end
• Unless a line is horizontal or verticalit
  intersects all four clipping planes
• Compute all four intersections

tbottom
y bottom
tleft
x left
x right
tleft (left x0) / (x1 x0) tright (right
x0) / (x1 x0) tbottom (bottom y0) / (y1
y0) ttop (top y0) / (y1 y0)
17
Parametric Clipping
(x1,y1)
tright
ttop
y top

• Cohen-Sutherland iterates through each edge
  intersection and computes them even if they are
  later clipped
• Liang-Barsky computes all at once postponing
  division to end
• Unless a line is horizontal or verticalit
  intersects all four clipping planes
• Compute oriented numerators and denominators of
  all four intersections
• Sign of denominator
• positive ? entering
• negative ? leaving

tbottom
y bottom
tleft
(x0,y0)
x left
x right
tleft (left x0) / (x1 x0) tright -(right
x0) / -(x1 x0) tbottom (bottom y0) /
(y1 y0) ttop -(top y0) / -(y1 y0)
18
Parametric Clipping
(x1,y1)
tright
ttop
y top

• Compute ordered list of t-intersections
• 0 lt tleft lt tbottom lt ttop lt tright lt 1

tbottom
y bottom
tleft
(x0,y0)
x left
x right
tleft (left x0) / (x1 x0) tright -(right
x0) / -(x1 x0) tbottom (bottom y0) /
(y1 y0) ttop -(top y0) / -(y1 y0)
19
Parametric Clipping
(x1,y1)
leave
leave
y top

• Compute ordered list of t-intersections
• 0 lt tleft lt tbottom lt ttop lt tright lt 1
• And corresponding orientations
• enter, enter, leave, leave

y bottom
enter
enter
(x0,y0)
x left
x right
tleft (left x0) / (x1 x0) tright -(right
x0) / -(x1 x0) tbottom (bottom y0) /
(y1 y0) ttop -(top y0) / -(y1 y0)
20
Parametric Clipping
(x1,y1)
leave
y top

• Compute ordered list of t-intersections
• 0 lt tleft lt tbottom lt ttop lt tright lt 1
• And corresponding orientations
• enter, enter, leave, leave
• Keep only the ltenter,leavegt pair

y bottom
enter
(x0,y0)
x left
x right
tleft (left x0) / (x1 x0) tright -(right
x0) / -(x1 x0) tbottom (bottom y0) /
(y1 y0) ttop -(top y0) / -(y1 y0)
21
Polygon Clipping
C
y top

• Sutherland-Hodgman
• Polygon ABC

y bottom
A
B
x left
x right
22
Polygon Clipping
C
y top

• Sutherland-Hodgman
• Polygon ABC
• Clip left A1BCA2

A2
y bottom
B
A
A1
x left
x right
23
Polygon Clipping
C
y top

• Sutherland-Hodgman
• Polygon ABC
• Clip left A1BCA2
• Clip right A1B1B2CA2

A2
B2
y bottom
B
B1
A1
x left
x right
24
Polygon Clipping
C
y top

• Sutherland-Hodgman
• Polygon ABC
• Clip left A1BCA2
• Clip right A1B1B2CA2
• Clip bottom A1B1B2CA2

A2
B2
y bottom
A1
B1
B1
A1
x left
x right
25
Polygon Clipping
C
C2
C1
y top

• Sutherland-Hodgman
• Polygon ABC
• Clip left A1BCA2
• Clip right A1B1B2CA2
• Clip bottom A1B1B2CA2
• Clip top A1B1B2C1C2A2

A2
B2
y bottom
A1
B1
B1
A1
x left
x right
26
Concave Clipping

• Sutherland-Hodgman
• Clip segments even if they are trivially
  rejectible (rejectionable?)
• Outputs a single polygon that appears as multiple
  polygons
• Reversed edges dont get filled

27
Clipping in 3-D (4-D)

• Need to keep depth (z-coordinate) of geometry for
  visible surface detection
• Generalize oriented screen edge to oriented
  clipping plane C (A,B,C,D)
• Then any homogeneous pointP (x,y,z,w)T
  classified as
• on if C P 0
• in if C P gt 0
• out if C P lt 0

(A,B,C)
Ax By Cz D 0 ?? wAx wBy wCz wD 0
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Graphics Pipeline
ModelCoords
ModelXform
WorldCoords
ViewingXform
ViewingCoords
StillClipCoords.
Clipping
ClipCoords.
Homogeneous Divide
Window Coordinates
Window to Viewport
Viewport Coordinates