This document discusses different techniques for clipping graphics objects to a viewing window, including: - Point and line clipping, where points and line segments outside the window are "clipped" and not drawn. - The Cohen-Sutherland and Liang-Barsky line clipping algorithms, which use outcodes and parametric equations to efficiently determine which portions of lines are inside the window. - Area clipping techniques like the Sutherland-Hodgman algorithm, which successively clips a polygon against each boundary of the window by comparing vertices and finding intersection points.

1. UNIT-3
Computer Graphics :
Viewing In 2D

2. Contents
 Windowing Concepts
 Clipping
 Introduction
 Brute Force
 Cohen-Sutherland Clipping Algorithm
 Area Clipping
 Sutherland-Hodgman Area Clipping Algorithm

3. Windowing I
 A scene is made up of a collection of objects
specified in world coordinates
World Coordinates

4. Windowing II
 When we display a scene only those objects within a
particular window are displayed
wymax
wymin
wxmin wxmax
Window
World Coordinates

5. Windowing III
 Because drawing things to a display takes time we
clip everything outside the window
wymax
wymin
wxmin wxmax
World Coordinates
Window

6. Clipping
 For the image below consider which lines and points
should be kept and which ones should be clipped
wymax
wymin
wxmin wxmax
Window
P1
P2
P3
P6
P5P7
P10
P9
P4
P8

7. Point Clipping
 Easy - a point (x,y) is not clipped if:
wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymax
 otherwise it is clipped
wymax
wymin
wxmin wxmax
Window
P1
P2
P5
P7
P10
P9
P4
P8
Clipped
Points Within the Window
are Not Clipped
Clipped
Clipped
Clipped

8. Line Clipping
 Harder - examine the end-points of each line to see if
they are in the window or not
Situation Solution Example
Both end-points inside
the window
Don’t clip
One end-point inside
the window, one
outside
Must clip
Both end-points
outside the window
Don’t know!

9. 9
Cohen-Sutherland Algorithm
 Idea: eliminate as many cases as possible
without computing intersections
 Start with four lines that determine the sides of
the clipping window
x = xmaxx = xmin
y = ymax
y = ymin

10. 10
The Cases
 Case 1: both endpoints of line segment inside
all four lines
 Draw (accept) line segment as is
 Case 2: both endpoints outside all lines and on
same side of a line
 Discard (reject) the line segment
x = xmaxx = xmin
y = ymax
y = ymin

11. 11
The Cases
 Case 3: One endpoint inside, one outside
 Must do at least one intersection
 Case 4: Both outside
 May have part inside
 Must do at least one intersection
x = xmaxx = xmin
y = ymax

12. 12
Defining Outcodes
 For each endpoint, define an outcode
 Outcodes divide space into 9 regions
 Computation of outcode requires at most 4
subtractions
b0b1b2b3
b0 = 1 if y > ymax, 0 otherwise
b1 = 1 if y < ymin, 0 otherwise
b2 = 1 if x > xmax, 0 otherwise
b3 = 1 if x < xmin, 0 otherwise

13. 13
Using Outcodes
 Consider the 5 cases below
 AB: outcode(A) = outcode(B) = 0
 Accept line segment

14. 14
Using Outcodes
 CD: outcode (C) = 0, outcode(D)  0
 Compute intersection
 Location of 1 in outcode(D) determines which edge
to intersect with
 Note if there were a segment from A to a point in a
region with 2 ones in outcode, we might have to do
two interesections

15. 15
Using Outcodes
 EF: outcode(E) logically ANDed with outcode(F)
(bitwise)  0
 Both outcodes have a 1 bit in the same place
 Line segment is outside of corresponding side of
clipping window
 reject

16. 16
Using Outcodes
 GH and IJ: same outcodes, neither zero but
logical AND yields zero
 Shorten line segment by intersecting with one of
sides of window
 Compute outcode of intersection (new endpoint of
shortened line segment)
 Reexecute algorithm

17. 17
Efficiency
 In many applications, the clipping window is small
relative to the size of the entire data base
 Most line segments are outside one or more side of
the window and can be eliminated based on their
outcodes
 Inefficiency when code has to be reexecuted for
line segments that must be shortened in more
than one step

18. Liang – Barsky clipping
 In computer graphics, the Liang–Barsky
algorithm (named after You-Dong
Liang and Brian A. Barsky) is a line
clipping algorithm.
 The Liang–Barsky algorithm uses the parametric
equation of a line and inequalities describing the
range of the clipping window to determine the
intersections between the line and the clipping
window. With these intersections it knows which
portion of the line should be drawn.
 This algorithm is significantly more efficient
than Cohen–Sutherland.

19.  The idea of the Liang-Barsky clipping algorithm is
to do as much testing as possible before
computing line intersections. Consider first the
usual parametric form of a straight line:
 A point is in the clip window, if

20. To compute the final line segment:
 A line parallel to a clipping window edge
has for that boundary.
 If for that the line is completely outside
and can be eliminated.
 When the line proceeds outside to inside
the clip window and when , the line
proceeds inside to outside.
 For nonzero , gives the intersection
point.

21.  Here

22. Liang-Barsky
Enter
Enter
Leave
Leave
Enter
Leave
Enter
Leave

23. Liang-Barsky - Algorithm
 Compute entering u values, which are qk/pk for each
pk<0
 Compute leaving u values, which are qk/pk for each
pk>0
 Parameter value for small u end of line is:usmall=
max(0, entering u’s)
 parameter value for large t end of line is: ularge=min(1,
leaving u’s)
 If usmall<ularge, there is a line segment - compute
endpoints by substituting t values
 Improvement (and actual Liang-Barsky):
 compute u’s for each edge in turn (some rejects occur earlier
like this.)

24. Area Clipping
 Similarly to lines, areas must
be clipped to a window
boundary
 Consideration must be taken
as to which portions of the
area must be clipped

25. Sutherland-Hodgman Area Clipping
Algorithm
 A technique for clipping areas
developed by Sutherland &
Hodgman
 Put simply the polygon is clipped
by comparing it against each
boundary in turn
Original Area Clip Left Clip Right Clip Top Clip Bottom
Sutherland
turns up
again. This
time with
Gary Hodgman with
whom he worked at
the first ever
graphics company
Evans & Sutherland

27.  The Sutherland-Hodgeman Polygon-Clipping
Algorithms clips a given polygon successively against
the edges of the given clip-rectangle.
 These clip edges are denoted with e1, e2, e3, and
e4, here.
 The closed polygon is represented by a list of its
vertices (v1 to vn; Since we got 15 vertices in the
example shown above, vn = v15.).
 Clipping is computed successively for each edge.
The output list of the previous clipping run is used as
the input list for the next clipping run.

36. Sutherland-Hodgman Area Clipping
Algorithm (cont…)
 To clip an area against an individual boundary:
 Consider each vertex in turn against the boundary
 Vertices inside the boundary are saved for clipping against
the next boundary
 Vertices outside the boundary are clipped
 If we proceed from a point inside the boundary to one
outside, the intersection of the line with the boundary is
saved
 If we cross from the outside to the inside intersection point
and the vertex are saved

37. Sutherland-Hodgman Example
 Each example shows
the point being
processed (P) and the
previous point (S)
 Saved points define
area clipped to the
boundary in question
S
P
Save Point P
S
P
Save Point I
I
P
S
No Points Saved
S
P
Save Points I & P
I

38. Other Area Clipping Concerns
 Clipping concave areas can be a little more tricky as
often superfluous lines must be removed
 Clipping curves requires more work
 For circles we must find the two intersection points on
the window boundary
Window WindowWindow Window