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

Clip Top

Vertices in Clip Right

Clip Bottom

Clip Left

Clipped vertices out

Inside-Outside Testing

• Edges/Planes store a vector pointing toward the outside of the clip region

• Dot products give inside/outside information

Outside Insiden

s

f

i

x

0

0

0

x)(fn

x)(in

x)(sn

s-x

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

ss

sss

sss

wz

wyw

wxw

0

Finding Intersection Pts Watt Sect 1.4.3, 1.4.4

• Use the parametric form for the line between two points, x1 and x2:

• For planes of the form x=a:

• Similar forms for y=a, z=a• Solution for general plane in textbook

10 )()( 121 ttt xxxx

))()(

)(),(

)(

)(,( 1

12

1211

12

121 xa

xx

zzzxa

xx

yyyai

x

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

4

1 2

3

4

3

4

1 2

3

4

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.

• Trivial reject: – outcode(x1)&outcode(x2)!=0

• Trivial accept:– outcode(x1)|outcode(x2)==0

• Which edges to clip against?– outcode(x1)^outcode(x2)

1 2

3

4

0010

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:

12max1min

12max1min

yyyyytyy

xxxxxtxx

Liang-Barsky (2)

• Infinite line intersects clip region edges when:

k

kk p

qt

p1 x q1 x1 x min

p2 x q2 x max x1

p3 y q3 y1 ymin

p4 y q4 ymax y1

where

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)

Enter Enter

LeaveLeave

Enter

Leave

EnterLeave

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

unsuitable1 2

3

4

Reject

TopTop, Right

Top, Bottom

Left

Left, bottom

Weiler Atherton Polygon Clipping

• For clockwise polygon:– for out-to-in pair, follow

usual rule

– for in-to-out pair, follow clip edge

• Easiest to start outside

Inside

1

2

3

4

5

6

7

8

go up

go up

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)

Changes to

Rearranging pointersmakes it possible to enumerate all componentsof the intersection