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Clipping
3D Rendering Pipeline 3D Geometric Primitives
Modeling Transformation
Viewing Transformation
Projection Transformation
Lighting
Image
Clipping
Scan Conversion Draw pixels (includes texturing, hidden surface, etc.)
Clip primitives outside camera’s view
Transform into 2D camera coordinate system
Transform into 3D camera coordinate system
Transform into 3D world coordinate system
Illuminate according to lighting and reflectance
2D Rendering Pipeline
Viewport Transformation
Scan Conversion
Clipping
2D Primitives
Image
Clip portions of geometric primitives
residing outside the window
Fill pixels representing primitives
in screen coordinates
Transform the clipped primitives
from screen to image coordinates
3D Primitives
Clipping
Avoid drawing parts of primitives outside window
Window defines part of scene being viewed
Must draw geometric primitives only inside window
Screen Coordinates
Window
Clipping
Avoid drawing parts of primitives outside window
Window defines part of scene being viewed
Must draw geometric primitives only inside window
Viewing
Window
Clipping
Avoid drawing parts of primitives outside window
Points
Lines
Polygons
Circles
etc.
Viewing
Window
Point Clipping
Is point (x,y) inside the clip window?
Window
wx1 wx2
wy2
wy1
(x,y)
inside =
(x >= wx1) &&
(x <= wx2) &&
(y >= wy1) &&
(y <= wy2);
Line Clipping
Find the part of a line inside the clip
window
P1
P10
P9
P8
P7
P4 P3
P6
P5
P2
Before Clipping
P’8
P’7
P4 P3
P6
P’5
After Clipping
Line Clipping
Find the part of a line inside the clip
window
Cohen Sutherland Line Clipping
Use simple tests to classify easy cases
first
P1
P10
P9
P8
P7
P4 P3
P6
P5
P2
Key Idea: Encode Regions
Bit
Value
0 1
xxmin x<xmin 1
xxmax x>xmax 2
yymin y<ymin 3
yymax y>ymax 4
Endpoint Code Classification
Assign code to end points of each line segment
line is inside if (c1 OR c2 ==0)
line is outside if )c1 AND c2 != 0)
Remove all such lines
Compute intersections with window boundary for all
lines that can’t be classified quickly
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P1
P10
P9
P8
P7
P4 P3
P6
P5
P2
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P1
P10
P9
P8
P7
P4 P3
P6
P5
P2
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P7
P4 P3
P6
P5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P7
P4 P3
P6
P5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P7
P4 P3
P6
P5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P7
P4 P3
P6
P5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P’7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P8
P’7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P’8
P’7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P’8
P’7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P’8
P’7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P’8
P’7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P’8
P’7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P10
P9
P’8
P’7
P4 P3
P6
P’5
Cohen Sutherland Line
Clipping
Bit 1 Bit 2
Bit 3
Bit 4
0000 0100 1000
0001 0101 1001
0010 0110 1010
P’8
P’7
P4 P3
P6
P’5
Intersecting An Edge and a Line
Denote p(t)=p0+(p1-p0)t t[0..1]
Let qi be a point on the edge Li with outside pointing normal Ni.
V(t) = p(t)-qi is a parameterized vector from qi to the segment p(t).
We are looking for t satisfying the equation: Ni· V)t) = 0
p0
p1
Ni
Li
p(t) V(t)
qi
Finding the Parameter t of the
Intersection Point
0 = Ni· V)t)
= Ni· )p)t)-qi)
= Ni· )p0+(p1-p0)t-qi)
= Ni· )p0-qi) + Ni· )p1-p0)t
p0
p1
Ni qi
Li
)( where
)(
)(
)(
01
0
01
0
pp
N
qpN
ppN
qpNt
i
ii
i
iii
p(ti)
V(ti)
t at intersection
Cyrus-Beck Line Clipping Example
p1
p0
PL
PE
p0
p1
PE
PE
PL
PL
p1
PL
PE p0
Line is outside
Cyrus-Beck Line Clipping
The intersection of p(t) with all four edges Li is computed,
resulting in up to four ti values per segment.
If ti<0 or ti>1 , ti can be discarded (why?).
Based on the sign of Ni· , each intersection point is classified
as PE (potentially entering) or PL (potentially leaving).
PE with the largest t and PL with the smallest t provide the
domain of p(t) inside W.
If PL comes before PE the line is outside!
Note: If Ni· =0, t has no solution. However, in this case V(t)
Ni and there is no defined intersections.
Polygon Clipping
Find the part of a polygon inside the clip
window? Before Clipping
Polygon Clipping
Find the part of a polygon inside the clip
window? After Clipping
Naïve Clipping
Find intersections
Naïve Clipping
Find intersections
Remove all outside vertices
Sutherland Hodgeman
Clipping
Clip to each window boundary one at a time
Sutherland Hodgeman
Clipping
Clip to each window boundary one at a time
Sutherland Hodgeman
Clipping
Clip to each window boundary one at a time
Sutherland Hodgeman
Clipping
Clip to each window boundary one at a time
Sutherland Hodgeman
Clipping
Clip to each window boundary one at a time
Clipping to a Boundary
Assumes you have vertices ordered!
Loop around polygon:
Do inside test for each vertex in polygon
sequence
When crossing window boundary insert
new point
Remove points outside window boundary
Clipping to a Boundary
Outside
Inside
Window
Boundary
P1 P2
P5
P4
P3
Clipping to a Boundary
Outside
Inside
Window
Boundary
P1 P2
P5
P4
P3
Clipping to a Boundary
Outside
Inside
Window
Boundary
P1 P2
P5
P4
P3
Clipping to a Boundary
Outside
Inside
Window
Boundary
P1 P2
P5
P4
P3
P’
Clipping to a Boundary
Outside
Inside
Window
Boundary
P1 P2
P5
P4
P3
P’
Clipping to a Boundary
Outside
Inside
Window
Boundary
P1 P2
P5
P4
P3
P’
Clipping to a Boundary
Outside
Inside
Window
Boundary
P1 P2
P5
P4
P3
P’ P’’
Clipping to a Boundary
Outside
Inside
Window
Boundary
P1 P2
P’ P’’
Relation Between Successive
Vertices
clip boundary
inside outside u
v
clip boundary
inside outside
u
v
clip boundary
inside outside v u w
u
clip boundary
inside outside
v w
v added to output list
w added to output list no output w and v added to
output list
Assume vertex u has been dealt with, vertex v follows:
Sutherland Hodgeman
Clipping
Assumes you have vertices ordered in List L
Initialize: Lin = L, Lout =
For each window edge e {
u = last(Lin)
Loop over v Lin {
If v is inside e then
if u is outside e
then Lout = Lout intersect(e,<u,v>)
Lout = Lout v
Else
if u is inside e
then Lout = Lout intersect(e,<u,v>)
u = v
}
Lin = Lout
}
e
Notes
Works for any convex clipping polygon
Good for drawing – not for modeling:
Overlapping edges
2D Rendering Pipeline
Viewport Transformation
Scan Conversion
Clipping
2D Primitives
Image
Clip portions of geometric primitives
residing outside the window
Fill pixels representing primitives
in screen coordinates
Transform the clipped primitives
from screen to image coordinates
3D Primitives
Viewport Transformation
Transform 2D geometric primitives from
screen coordinate system (normalized device
coordinates) to image coordinate system (pixels)
Image
Screen
Viewport
Window
Window-to-Viewport Mapping
vx1 vx2 vy1
vy2
wx1 wx2 wy1
wy2 Window Viewport
Screen Coordinates Image Coordinates
(wx,wy) (vx,vy)
)(
)(
1
12
12
1
1
12
121
yy
yy
yy
yy
xx
xx
xxxx
wwww
vvvv
wwww
vvvv
Summary
Viewport Transformation
Scan Conversion
Clipping
2D Primitives
Image
Clip portions of geometric primitives
residing outside the window
Fill pixels representing primitives
in screen coordinates
Transform the clipped primitives
from screen to image coordinates
3D Primitives
Summary of Transformations
Modeling Transformation
Viewing Transformation
2D Image Coordinates
Projection Transformation
Window-to-Viewport Transformation
3D Object Coordinates
3D World Coordinates
3D Camera Coordinates
2D Screen Coordinates
p(x,y,z)
p’)x’,y’)
Viewing transformations
Modeling transformation
Viewport transformation
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