Computer Graphics :
Viewing In 2D
UNIT-3
Contents
 Windowing Concepts
 Clipping
 Introduction
 Brute Force
 Cohen-Sutherland Clipping Algorithm
 Area Clipping
 Sutherland-Hodgman Area Clipping Algorithm
Windowing I
 A scene is made up of a collection of objects specified in world
coordinates
World Coordinates
Windowing II
 When we display a scene only those objects within a particular
window are displayed
Window
wymax
wymin
wxmax
wxmin
World Coordinates
Windowing III
 Because drawing things to a display takes time we clip everything
outside the window
Window
wymax
wymin
wxmax
wxmin
World Coordinates
Clipping
 For the image below consider which lines and points should be
kept and which ones should be clipped
P4
Window
wymax
P2
P6
P3
P1
P5
P7
P9
P8
wymin
P10
wxmin
wxmax
Point Clipping
 Easy - a point (x,y) is not clipped if:

wxmin ≤ x ≤ wxmax AND wymin ≤ y ≤ wymax
 otherwise it is clipped
P4 Clipped
Clipped
Window
wymax
Clipped
P7
P5
P2
P1
Points Within the Window
are Not Clipped
P9
P8
wymin
Clipped
wxmin
P10
wxmax
Line Clipping
 Harder - examine the end-points of each line to see if they are in
the window or not
Situation
Solution
Both end-points inside
Don’t clip
the window
One end-point inside
the window, one
outside
Must clip
Both end-points
outside the window
Don’t know!
Example
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
y = ymax
x = xmin
x = xmax
y = ymin
9
The Cases
 Case 1: both endpoints of line segment inside all four lines
 Draw (accept) line segment as is
y = ymax
x = xmin
x = xmax
y = ymin
 Case 2: both endpoints outside all lines and on same side of
a line
 Discard (reject) the line segment
10
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
y = ymax
x = xmin
x = xmax
11
Defining Outcodes
 For each endpoint, define an outcode
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
 Outcodes divide space into 9 regions
 Computation of outcode requires at most 4 subtractions
12
Using Outcodes
 Consider the 5 cases below
 AB: outcode(A) = outcode(B) = 0
 Accept line segment
13
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
14
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
15
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
16
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
17
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.
 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




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.
 Here
Liang-Barsky
Leave
Enter
Leave
Leave
Leave
Enter
Enter
Enter
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.)
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
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
Sutherland
turns up
again. This
time with
Gary Hodgman with
whom he worked at
the first ever
graphics company
Evans & Sutherland
Clip Top
Clip Bottom
 The Sutherland-Hodgeman Polygon-Clipping Algorithms clips a
given polygon successively against the edges of the given cliprectangle.
 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.
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
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
S
P
I
P
Save Point P
Save Point I
S
P
I
P
S
No Points Saved
Save Points I & P
Other Area Clipping Concerns
 Clipping concave areas can be a little more tricky as often
superfluous lines must be removed
Window
Window
Window
 Clipping curves requires more work
Window
 For circles we must find the two intersection points on the window
boundary