Clipping
• Apart from clipping to the view volume, clipping
is a basic operation in many other algorithms
– Breaking space up into chunks
– 2D drawing and windowing
– Modelling
• May require more complex geometry than
rectangular boxes
Hardware Sutherland-Hodgman
• Suitable for hardware implementation
– Only need the clip edge, the endpoints of the current
edge, and the last output point
– Polygon edges are output as they are found, and passed
right on to the next clip region edge
Vertices in
Clip
Top
Clip
Right
Clipped vertices out
Clip
Bottom
Clip
Left
Inside-Outside Testing
• Edges/Planes store a vector pointing toward the
outside of the clip region
• Dot products give inside/outside information
Outside
n
Inside
n  (s  x)  0
x
n  (i  x)  0
n  (f  x)  0
f
s-x
i
s
Inside/Outside in Screen Space
• In screen space, clip planes are xs=±1, ys=±1,
zs=0, zs=1
• Inside/Outside reduces to comparisons before
perspective divide
 ws  xs  ws
 ws  ys  ws
0  zs  ws
Finding Intersection Pts Watt Sect 1.4.3, 1.4.4
• Use the parametric form for the line between two
points, x1 and x2:
x(t )  x1  (x2  x1 )t
0  t 1
• For planes of the form x=a:
( y2  y1 )
( z2  z1 )
xi  (a, y1 
(a  x1 ), z1 
(a  x1 ))
( x2  x1 )
( x2  x1 )
• Similar forms for y=a, z=a
• Solution for general plane in textbook
Additional Clipping Planes
• Useful for doing things like cut-away views
– Use a clip plane to cut off part of the object
– Only works if piece to be left behind is convex
• OpenGL allows you to do it
• Also one way to use OpenGL to identify objects in
a region of space (uses the selection mechanism)
Clipping Lines
• Lines can also be clipped by Sutherland-Hodgman
– Slow, unless you already have hardware
• Better algorithms exist
– Cohen-Sutherland
– Liang-Barsky
– Nicholl-Lee-Nicholl
Cohen-Sutherland (1)
• Works basically the same as Sutherland-Hodgman
– Was developed earlier
• Clip line against each edge of clip region in turn
– If both endpoints outside, discard line and stop
– If both endpoints in, continue to next edge (or finish)
– If one in, one out, chop line at crossing pt and continue
Cohen-Sutherland (2)
1
2
3
3
4
4
3
3
4
4
1
2
1
2
Cohen-Sutherland (3)
• Some cases lead to premature acceptance or
rejection
– If both endpoints are inside all edges
– If both endpoints are outside one edge
Cohen-Sutherland - Details
• Only need to clip line against edges where one endpoint is out
• Use outcode to record endpoint in/out wrt each edge. One bit
per edge, 1 if out, 0 if in.
1
2
• Trivial reject:
0010
– outcode(x1)&outcode(x2)!=0
• Trivial accept:
3
– outcode(x1)|outcode(x2)==0
• Which edges to clip against?
– outcode(x1)^outcode(x2)
4
0101
Liang-Barsky Clipping
• Parametric clipping - view line in parametric form
and reason about the parameter values
• More efficient, as not computing the coordinate
values at irrelevant vertices
• Clipping conditions on parameter: Line is inside
clip region for values of t such that:
xmin  x1  tx  xmax
ymin  y1  ty  ymax
x  x2  x1
y  y2  y1
Liang-Barsky (2)
• Infinite line intersects clip region edges when:
qk
tk 
pk
p1  x q1  x 1  x mi n
where
p 2  x
q 2  x max  x 1
p 3  y q 3  y1  y mi n
p 4  y
q 4  y max  y1
Liang-Barsky (3)
• When pk<0, as t increases line goes from outside to
inside - enter
• When pk>0, line goes from inside to outside - leave
• When pk=0, line is parallel to an edge (clipping is
easy)
• If there is a segment of the line inside the clip region,
sequence of infinite line intersections must go: enter,
enter, leave, leave
Liang-Barsky (4)
Leave
Enter
Leave
Leave
Leave
Enter
Enter
Enter
Liang-Barsky - Algorithm
• Compute entering t values, which are qk/pk for each pk<0
• Compute leaving t values, which are qk/pk for each pk>0
• Parameter value for small t end of line is:tsmall= max(0,
entering t’s)
• parameter value for large t end of line is: tlarge=min(1,
leaving t’s)
• if tsmall<tlarge, there is a line segment - compute endpoints
by substituting t values
• Improvement (and actual Liang-Barsky):
– compute t’s for each edge in turn (some rejects occur earlier like
this.)
Nicholl-Lee-Nicholl clipping
• Some edges are irrelevant to
clipping, particularly if one
vertex lies inside region.
• Cases:
– x1 in
– x1 in corner region
– x1 in edge region
• For each case, we generate
specialized test regions for x2,
which use simple tests (slope,
>, <), and tell which edges to
clip against.
a
a
a
Nicholl-Lee-Nicholl (2)
• Special cases for each endpoint location and slope
• Number of cases explodes in 3D, making it
unsuitable
1
Reject
2
3
Top
Left
Top, Right
4
Left, bottom
Top, Bottom
Weiler Atherton Polygon Clipping
Inside
• For clockwise polygon:
– for out-to-in pair, follow
usual rule
– for in-to-out pair, follow
clip edge
2
3
go up
• Easiest to start outside
1
4
5
6
8
go up
7
General Clipping
• Clipping general against general polygons is quite hard
• Outline of Weiler algorithm:
–
–
–
–
Replace crossing points with vertices
Double all edges and form linked lists of edges
Change links at vertices
Enumerate polygon patches
• Can use clipping to break concave polygon into convex
pieces; main issue is inside-outside for edges
Weiler Algorithm (1)
Rearranging pointers
makes it possible to
enumerate all components
of the intersection
Changes to