CS 455 Computer Graphics

1
CS 455 Computer Graphics

• Clipping

2
Line Clipping

• What happens when one or both endpoints of a
  line segment are not inside the specified drawing
  area?
• Draw just the portions of a line (or object)
  that fall within a given region/window/screen
  (usually rectangular)

Drawing Area
3
Line Clipping

• Strategies for clipping
• Check (in inner loop) if each point is inside ?
  Works, but slow
• Clip invalid coordinate(s) to boundary ?
  Incorrect results
• Find intersection of line with boundary ? Correct

if (x gt xmin x lt xmax y gt ymin y
lt ymax) drawPoint(x,y,c)
if (x lt xmin) x xmin else if (x gt xmax) x
xmax if (y lt ymin) y ymin else if (y gt
ymax) y ymax
Clip x
Output
Input
Clip y
Clip line to intersection
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Line Clipping Possible Configurations

• Both endpoints are inside the region (line AB)
• No clipping necessary
• One endpoint in, one
• out (line CD)
• Clip at intersection point
• Both endpoints outside
• the region
• No intersection (lines EF, GH)
• Line intersects the region (line IJ)
• Clip line at both intersection points

J
D
I
A
F
C
H
B
E
G
5
Line Clipping Cohen-Sutherland

• Basic algorithm
• Accept (and draw) lines that have both endpoints
  inside the region

• Reject (and dont draw) lines that have both
  endpoints less than xmin or ymin or greater than
  xmax or ymax

• Clip the remaining lines at a region boundary and
  repeat steps 1 and 2 on the clipped line segments

6
Cohen-Sutherland Accept/Reject Tests

• Assign 4-bit code to each endpoint corresponding
  to its position relative to region
• First bit (1000) if y gt ymax
• Second bit (0100) if y lt ymin
• Third bit (0010) if x gt xmax
• Fourth bit (0001) if x lt xmin
• Test
• if code0 OR code1 0000
• accept (draw)
• else if code0 AND code1 ? 0000
• reject (dont draw)
• else clip and retest

1000
1001
1010
0001
0010
0000
0100
0101
0110
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Cohen-Sutherland Accept/Reject

• Accept/reject/redo all based on bit-wise Boolean
  ops.

1000
1001
1010
0001
0010
0000
0100
0101
0110
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Cohen-Sutherland Line Clipping

• Intersection algorithm
• if code0 ? 0000 then code code0
• else code code1
• dx x1 x0 dy y1 y0
• if code AND 1000 then begin // ymax
• x x0 dx (ymax y0) / dy y ymax
• end
• else if code AND 0100 then begin // ymin
• x x0 dx (ymin y0) / dy y ymin
• end
• else if code AND 0010 then begin // xmax
• y y0 dy (xmax x0) / dx x xmax
• end
• else begin // xmin
• y y0 dy (xmin x0) / dx x xmin
• end
• if code code0 then begin x0 x y0 y end
• else begin x1 x y1 y end

(x1, y1)
(x, y)
ymax
dy
(x0, y0)
dx
ymin
xmin
xmax
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Cohen-Sutherland Line Clipping Example

• Intersection algorithm
• if code0 ? 0000 then code code0
• else code code1
• dx x1 x0 dy y1 y0
• if code AND 1000 then begin // ymax
• x x0 dx (ymax y0) / dy y ymax
• end
• else if code AND 0100 then begin // ymin
• x x0 dx (ymin y0) / dy y ymin
• end
• else if code AND 0010 then begin // xmax
• y y0 dy (xmax x0) / dx x xmax
• end
• else begin // xmin
• y y0 dy (xmin x0) / dx x xmin
• end
• if code code0 then begin x0 x y0 y end
• else begin x1 x y1 y end

Code
dx
y
x
dy
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Cohen-Sutherland Line Clipping Example

• Intersection algorithm
• if code0 ? 0000 then code code0
• else code code1
• dx x1 x0 dy y1 y0
• if code AND 1000 then begin // ymax
• x x0 dx (ymax y0) / dy y ymax
• end
• else if code AND 0100 then begin // ymin
• x x0 dx (ymin y0) / dy y ymin
• end
• else if code AND 0010 then begin // xmax
• y y0 dy (xmax x0) / dx x xmax
• end
• else begin // xmin
• y y0 dy (xmin x0) / dx x xmin
• end
• if code code0 then begin x0 x y0 y end
• else begin x1 x y1 y end

Code
dx
y
x
dy
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Cohen-Sutherland Line Clipping Example

• Intersection algorithm
• if code0 ? 0000 then code code0
• else code code1
• dx x1 x0 dy y1 y0
• if code AND 1000 then begin // ymax
• x x0 dx (ymax y0) / dy y ymax
• end
• else if code AND 0100 then begin // ymin
• x x0 dx (ymin y0) / dy y ymin
• end
• else if code AND 0010 then begin // xmax
• y y0 dy (xmax x0) / dx x xmax
• end
• else begin // xmin
• y y0 dy (xmin x0) / dx x xmin
• end
• if code code0 then begin x0 x y0 y end
• else begin x1 x y1 y end

Code 1010
dx
y
x
dy
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Cohen-Sutherland Line Clipping Example

• Intersection algorithm
• if code0 ? 0000 then code code0
• else code code1
• dx x1 x0 dy y1 y0
• if code AND 1000 then begin // ymax
• x x0 dx (ymax y0) / dy y ymax
• end
• else if code AND 0100 then begin // ymin
• x x0 dx (ymin y0) / dy y ymin
• end
• else if code AND 0010 then begin // xmax
• y y0 dy (xmax x0) / dx x xmax
• end
• else begin // xmin
• y y0 dy (xmin x0) / dx x xmin
• end
• if code code0 then begin x0 x y0 y end
• else begin x1 x y1 y end

Code 1010
dx 250
y
x
dy 150
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Cohen-Sutherland Line Clipping Example

• Intersection algorithm
• if code0 ? 0000 then code code0
• else code code1
• dx x1 x0 dy y1 y0
• if code AND 1000 then begin // ymax
• x x0 dx (ymax y0) / dy y ymax
• end
• else if code AND 0100 then begin // ymin
• x x0 dx (ymin y0) / dy y ymin
• end
• else if code AND 0010 then begin // xmax
• y y0 dy (xmax x0) / dx x xmax
• end
• else begin // xmin
• y y0 dy (xmin x0) / dx x xmin
• end
• if code code0 then begin x0 x y0 y end
• else begin x1 x y1 y end

Code 1010
dx 250
y
x
dy 150
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Cohen-Sutherland Line Clipping Example

• Intersection algorithm
• if code0 ? 0000 then code code0
• else code code1
• dx x1 x0 dy y1 y0
• if code AND 1000 then begin // ymax
• x x0 dx (ymax y0) / dy y ymax
• end
• else if code AND 0100 then begin // ymin
• x x0 dx (ymin y0) / dy y ymin
• end
• else if code AND 0010 then begin // xmax
• y y0 dy (xmax x0) / dx x xmax
• end
• else begin // xmin
• y y0 dy (xmin x0) / dx x xmin
• end
• if code code0 then begin x0 x y0 y end
• else begin x1 x y1 y end

Code 1010
dx 250
y 200
x 233
dy 150
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Cohen-Sutherland Line Clipping Example

• Intersection algorithm
• if code0 ? 0000 then code code0
• else code code1
• dx x1 x0 dy y1 y0
• if code AND 1000 then begin // ymax
• x x0 dx (ymax y0) / dy y ymax
• end
• else if code AND 0100 then begin // ymin
• x x0 dx (ymin y0) / dy y ymin
• end
• else if code AND 0010 then begin // xmax
• y y0 dy (xmax x0) / dx x xmax
• end
• else begin // xmin
• y y0 dy (xmin x0) / dx x xmin
• end
• if code code0 then begin x0 x y0 y end
• else begin x1 x y1 y end

Code 1010
dx 250
y 200
x 233
dy 150
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Cohen-Sutherland Line Clipping Example

• Intersection algorithm
• if code0 ? 0000 then code code0
• else code code1
• dx x1 x0 dy y1 y0
• if code AND 1000 then begin // ymax
• x x0 dx (ymax y0) / dy y ymax
• end
• else if code AND 0100 then begin // ymin
• x x0 dx (ymin y0) / dy y ymin
• end
• else if code AND 0010 then begin // xmax
• y y0 dy (xmax x0) / dx x xmax
• end
• else begin // xmin
• y y0 dy (xmin x0) / dx x xmin
• end
• if code code0 then begin x0 x y0 y end
• else begin x1 x y1 y end

Code 1010
dx 250
y 200
x 233
dy 150
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Cohen-Sutherland Line Clipping Summary

• Choose an endpoint outside the clipping region
• Using a consistent ordering (top to bottom, left
  to right) find a clipping border the line
  intersects
• Discard the portion of the line from the endpoint
  to the intersection point
• Set the new line to have as endpoints the new
  intersection point and the other original
  endpoint
• You may need to run this several times on a
  single line (e.g., a line that crosses multiple
  clip boundaries)

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Cohen-Sutherland Line Clip Examples
H
1000
1001
1010
F
D
G
0001
0000
C
0010
J
A
E
I
0101
0100
0110
B
A 0001 B 0100 OR 0101 AND
0000 subdivide
C 0000 D 0010 OR 0010 AND
0000 subdivide
E 0000 F 0000 OR 0000 AND
0000 accept
G 0000 H 1010 OR 1010 AND
0000 subdivide
I 0110 J 0010 OR 0110 AND
0010 reject
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Cohen-Sutherland Line Clip Examples
H
1000
1001
1010
G
D
G
0001
0000
C
C
0010
A
A
0101
0100
0110
B
A 0001 A 0001 remove
A 0001 B 0100 OR 0101 AND
0000 subdivide
C 0000 C 0000 OR 0000 AND 0000 accept
C 0000 D 1010 remove
G 0000 G 0000 OR 0000 AND
0000 accept
G 0000 H 1010 remove
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Cohen-Sutherland Line Clip Examples
1000
1001
1010
0001
0000
0010
A
B
0101
0100
0110
B
A 0001 B 0100 remove
B 0100 B 0100 OR 0100 AND
0100 reject
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Shadow maps
From http//en.wikipedia.org/wiki/Shadow_mapping
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Project 2
Perspective
Parallel (and orthonormal)
23
Polygon Clipping

• What about polygons?
• For concave polygons, the intersection
• with the clipping region may be complex

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Polygon Clipping Algorithm

• Clip polygon to ymin and ymax
• Create empty output vertex list (listout empty)
• Process input list (listin (v0, v1, , vn)
  where v0 vn) in order
• For each input vertex (vi where 0 ? i ? n1)
• If vi is inside region ? Add vi to end of listout
• If the line between vi and vi1 intersects
  specified boundaries ? Add intersection point(s)
  to end of listout
• Repeat clipping to xmin and xmax
• Post-process
• Find degenerate sections where both sides of
  polygon has collapsed to region boundary
• Remove those sections ? Create new polygon

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Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v0
ymin
inside region no
v9
v5
v1
v2
add p0 to output list
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0)
26
Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v1
ymin
inside region yes
v9
v5
v1
v2
add v1 to output list
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0, v1)
(p0)
27
Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v2
ymin
inside region yes
v9
v5
v1
v2
add v2, p1 to output list
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0, v1, v2, p1)
(p0, v1)
28
Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v3
ymin
inside region no
v9
v5
v1
v2
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0, v1, v2, p1)
29
Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v4
ymin
inside region no
v9
v5
v1
v2
add p2 to output list
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0, v1, v2, p1, p2)
(p0, v1, v2, p1)
30
Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v5
ymin
inside region yes
v9
v5
v1
v2
add v5, p3 to output list
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0, v1, v2, p1, p2, v5, p3)
(p0, v1, v2, p1, p2)
31
Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v6
ymin
inside region no
v9
v5
v1
v2
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0, v1, v2, p1, p2, v5, p3)
32
Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v7
ymin
inside region no
v9
v5
v1
v2
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0, v1, v2, p1, p2, v5, p3)
33
Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v8
ymin
inside region no
v9
v5
v1
v2
add p4 to output list
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0, v1, v2, p1, p2, v5, p3, p4)
(p0, v1, v2, p1, p2, v5, p3)
34
Polygon Clipping Example
v6
v7
Clip first to ymin and ymax
v0
v8
vertex v9
ymin
inside region yes
v9
v5
v1
v2
add v9, p5 to output list
ymax
v3
v4
Input vertex list (v0, v1, v2, v3, v4, v5, v6,
v7, v8, v9)
Output vertex list
(p0, v1, v2, p1, p2, v5, p3, p4, v9, p5)
(p0, v1, v2, p1, p2, v5, p3, p4)
35
Polygon Clipping Example
This gives us a new polygon
ymin
v9
v5
v1
v2
ymax
with vertices
(p0, v1, v2, p1, p2, v5, p3, p4, v9, p5)
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Polygon Clipping Example (cont.)
Now clip to xmin and xmax
xmin
xmax
p9
v9
p6
v5
p7
v1
v2
p8
Input vertex list (p0, v1, v2, p1, p2, v5,
p3, p4, v9, p5) Output vertex list (p0, p6, p7,
v2, p1, p8, p9, p3, p4, v9, p5)
37
Polygon Clipping Example (cont.)
Now post-process
xmin
xmax
p3
p4
p5
p0
p9
v9
p6
p7
v2
p1
v3
p8
Output vertex list (p0, p6, p7, v2,
p1, p8, p9, p3, p4, v9, p5) Post-process (p0,
p6, p9, p3,) and (p7, v2, p1, p8) and (v4, v9, p5)