2D Clipping

1
2D Clipping

• CS 4451

2
Motivation

• Programmers describe graphics primitives
• The graphics library
• scan converts or rasterizes the primitive
• primitive is also clipped to the clip rectangle
• Graphics primitives are displayed on the screen

3
What is this clip rectangle?

• By default, the clip rectangle is the entire
  canvas
• not necessary to draw outside the canvas
• for some devices, it is damaging (plotters)
• Sometimes it is convenient to restrict the clip
  rectangle to a smaller portion of the canvas
• partial canvas redraw for menus, dialog boxes,
  other obscuration

4
Sophisticated Clipping Regions

• Postscript and similar graphics APIs allow
  multiple clipping regions of arbitrary shape
• Seen commercially in printers,
• also NeXT DisplayPostscript
• Mac OS X DisplayPDF
• NeWS, an early competitor to X

5
Efficiency

• Just as with scan-conversion algorithms, clipping
  algorithms are called whenever primitives are
  rendered.
• Important to do a little extra thinking to
  increase efficiency.
• Moores Law is not a license for laziness

6
How to do Clipping

• Clip it before you scan convert it
• used mostly for lines, rectangles, and polygons,
  where clipping algorithms are simple and
  efficient
• analytical clipping

7
How to do Clipping (cont.)

• Clip it during/after you scan convert it
• scissoring, a brute force technique
• scan convert the primitive, only write pixels if
  inside the clipping region
• easy for thick and filled primitives as part of
  scan line fill
• only outside boundary of primitive need be
  examined
• if primitive is not much larger than much larger
  than clip region, most pixels will fall inside
• can be more efficient than analytical clipping

8
How to do Clipping (cont.)

• Clip it after you scan convert it
• render everything onto a temporary canvas and
  copy the clipping region
• wasteful, but simple and easy,
• often used for text

9
How to Choose

• Foley and van Dam suggest the following
• for floating point graphics libraries, try to use
  analytical clipping
• for integer graphics libraries
• analytical clipping for lines and polygons
• others, do during scan conversion
• sometimes both analytical and raster clipping
  performed

10
Analytical Line Clipping

• First consider endpoints
• xminltxltxmax
• yminltyltymax

ymax
ymin
xmin
xmax
11
Analytical Line Clipping

• Both endpoints inside, line trivially accepted
• One in and one out, line is partially inside
• Both outside, might be partially inside
• How to handle non-trivial cases?

ymax
ymin
xmin
xmax
12
Simultaneous Equations

• Determine if there are line intersections with
  clipping region edges
• Use parametric equations
• xx0t(x1-x0)
• yy0t(y1-y0)
• Slope interface form not useful for line segments

• Intersection if both t values in 0,1
• We can do better

ymax
ymin
xmin
xmax
13
Cohen-Sutherland

• First, check endpoints for trivial acceptance
• perform region checks for trivial rejection
• subdivide lines so one segment can be trivially
  rejected
• continued until remainder can be trivially
  accepted or rejected
• performs well for common cases
• accepting most everything, large clip region
• rejecting most everything, small clip region

14
Outcodes

• First Bit
• ygtymax sign of ymax-y
• Second Bit
• yltymin sign of y-ymin
• Third Bit
• xgtxmax sign of xmax-x
• Fourth Bit
• xltxmin sign of x-xmin
• If logical AND of endpointsis not zero, trivial
  reject,
• else, subdivide, using consistent
• edge order (here top to bottom, left to right)

1000
1001
1010
ymax
0000
0001
0010
ymin
0100
0101
0110
xmin
xmax
15
Other Algorithms

• Cohen-Sutherland sometimes performs a lot of
  fruitless clipping due to external intersections,
  but oldest, widely published, most common
• Nicholl, Lee, and Nicholl (1987) algorithm
  subdivides plane into more regions.
• Cyrus-Beck (1978) and Liang-Barsky (1984) are
  more efficient

16
Cyrus-Beck/Liang-Barsky

• C-B is a parametic line-clipping algorithm
• can be used for 2D line clipping against
  arbitrary convex polygons
• or 3D lines against arbitrary convex polyhedron
• L-B adds efficient trivial rejection tests

17
C-B Overview

• Using parametric equations, compute line segment
  intersections (actually, just values of t) with
  clipping region edges
• Determine if the four values of t actually
  correspond to real intersections
• Then calculate x and y values of the
  intersections
• Faster than Cohen-Sutherland, does not need to
  iterate
• L-B examines values of t for earlier reject

18
Parametric Intersection

• Equation of line segment
• P(t)P0 (P1 - P0)t
• Perpendicular
• NiP(t)-PEi0
• NiP0(P1-P0)t-PEi0
• NiP0-PEi NiP1-P0t0
• Let DP1-P0
• tNiP0-PEi/-NiD

Ei
PEi
P1
NiP(t)-PEilt0
NiP(t)-PEi0
NiP(t)-PEigt0
P0
Ni
19
Parametric Intersection

• tNiP0-PEi/-NiD
• denominator must be non-zero
• Ni?0 normal is not zero (mistake)
• D?0 line is not a single point
• NiD?0 edge and line not parallel

20
Classifying Intersections

• Each line segment will have four values of t, one
  for each edge
• Can we just chose values of t in 0,1?
• Can we chose intermediate intersections?
• We must classify

21
Classifying Intersections

• PE-Potentially Entering
• PL-Potentially Leaving
• can be classified by angle between P0P1 and Ni,
  sign of NiD
• PL if lt90, NiDgt0
• PE if gt90, NiDlt0
• Calculation done while calculating t

PL
PE
P0 is to the left in this example
22
Classifying Intersections

• PE-Potentially Entering
• PL-Potentially Leaving
• must find an appropriate (te, tl)
• if tegttl, then reject
• otherwise chose highest te and lowest tl

PL
PE
P0 is to the left in this example
23
Line Clipping Conclusion

• For upright (axis aligned) clipping rectangles,
  further simplifications possible
• L-B compares new te and tl to old tl or te values
  as they are computed for trivial rejection
• L-B also can reject parallel segments outside the
  clipping region

24
Line Clipping Conclusion

• Cohen-Sutherland
• most common, uses outcodes, loops on some
  segments
• Cyrus-Beck
• parametric, generalizes to 3D, no looping
• Liang-Barsky
• rejection, etc. enhancements to C-B
• Nicholl, et al.
• Fastest, but doesnt generalize to 3D

25
Clipping Circles

• Test against circles extent (bounding box)
• Test against each quadrants extent
• Optionally test octants
• Then analytically solve for arc and edge
  intersections
• Or
• If the circle is not large, do an extent test
• Use scissoring
• Clip each filled scan line

26
Clipping Ellipses

• Test against circles extent (bounding box)
• Test against each quadrants extent
• Then analytically solve for arc and edge
  intersections
• Or
• If the ellipse is not large, do an extent test
• Use scissoring
• Clip each filled scan line

27
Clipping Polygons
28
Clipping Characters