WINDOWING AND CLIPPING
POINTS TO COVER
Clipping
 Line clipping
 Polygon clipping
 Normalized transformation
 Viewing transformation
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CLIPPING
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Clipping refers to the removal of part of a scene.
Any procedure which identifies that portion of a picture which is
either inside or outside a region is referred to as a clipping
algorithm or clipping.
The region against which an object is to be clipped is called clipping
window. Usually it is rectangular in shape.
Window : it is the selected area of the picture.
Following are the graphics primitives that we are going to study
under clipping:
 point clipping
 line clipping
 polygon clipping
POINT CLIPPING
Display P = (x, y) if
xwmin <= x < = xwmax
ywmin < = y <= ywmax
LINE CLIPPING
Before clipping
After clipping
LINE CLIPPING
COHEN SUTHERLAND LINE CLIPPING
ALGORITHM
COHEN-SUTHERLAND: WORLD DIVISION
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Space is divided into regions based on the window
boundaries
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Each region has a unique four bit region code
Region codes indicate the position of the regions with respect to
the window
1001
3
2
1
above below right
Region Code Legend
0
0001
left
0101
1000
0000
Window
0100
1010
0010
0110
COHEN-SUTHERLAND ALGORITHM
1001
1000
1010
(Xmax,ymax)
bit 0 : x  xmin
bit 1 : x  xmax
0001
0000
0010
0100
0110
bit 2 : y  ymin
bit 3 : y  ymax
(Xmin,ymin)
0101
clip
rectangle
Region outcodes
COHEN-SUTHERLAND: LABELLING
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Every end-point of the line is labelled with the
appropriate region code
Three cases :
Case1: Lines inside the window : not clipped
Case2: lines outside the window : Take AND operation of
the outcodes of the end points of the line. If answer of AND
operation is non zero means line is outside the window and
hence discard those lines.
Case3 : lines partially inside the window :Lines for which
the AND operation of the end coordinates is ZERO . These
are identified as partially visible lines.
Case3: Other Lines
These lines are processed as follows:
Find the intersection of those lines with the edges of window.
 We can use the region codes to determine which window boundaries should
be considered for intersection.eg: region code 1000 then line intersects with
TOP boundary. Find intersection with that boundary only.
 Check other end of the line to get other intersection.
 Draw line between calculated intersection.
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Intersections are calculated as
y coordinate of the intersection point with a vertical boundary can be
calculated as
y = y1 + m(x – x1),
x = xmin or xmax and m = ( y2 – y1) / ( x2 – x1)
x coordinate of the intersection point with a horizontal boundary can
be calculated as
x = x1 + (y – y1) / m , y = ymin or ymax and m = ( y2 – y1) / ( x2 – x1)
ALGORITHM
1.
2.
3.
Read two endpoints of the line say P1 (x1,y1)
and p2(x2,y2)
Read two corners of the window (left –top ) and
(right-bottom) say (Wx1,Wy1) and (Wx2, Wy2)
Assign the region codes to the end co-or of P1.
1.
2.
3.
4.
4.
Set Bit 1 – if (x<xwmin)
Set Bit 2 – if (x >xwmax)
Set Bit 3 – if (y<ywmin)
Set Bit 4 – if (y<ywmax)
Apply visibility check on the line.
ALGORITHM CONTI…
5. If line is partially visible then1.
2.
If region codes for the both end points are non-zero , find
intersection points p1’ and p2’ with boundary edges by
checking the region codes.
If region code for any one end point is non-zero then find
intersection point p1’ or p2’ with the boundary edge.
6. Divide the line segments considering the
intersection points.
7. Draw the line segment
8. Stop.
POLYGON CLIPPING
Issues –
Edges of polygon need to be tested against clipping rectangle
May need to add new edges
Edges discarded or divided
Multiple polygons can result from a single polygon
SUTHERLAND-HODGMAN POLYGON CLIPPING
ALGORITHM
SUTHERLAND-HODGMAN POLYGON 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.
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Original Area
Clip Left
Clip Right
Clip Top
Clip Bottom
SUTHERLAND-HODGEMAN CLIPPING
Input/output for algorithm:
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Input: list of polygon vertices in order.
Output: list of clipped polygon vertices consisting of old
vertices (maybe) and new vertices (maybe)
Sutherland-Hodgman does four tests on every edge of
the polygon:
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Inside – Inside ( I-I)
Inside –Outside (I-O)
Outside –Outside (O-O)
Outside-Inside(O-I)
Output co-ordinate list is created by doing these tests
on every edge of poly.
SUTHERLAND-HODGEMAN CLIPPING
Edge from S to P takes one of four cases:
inside
outside
inside
outside
inside
outside
p
s
p
Save p
p
s
Save I
inside
p
s
I
Save nothing
Save I
and P
outside
s
SUTHERLAND-HODGEMAN CLIPPING
Four cases:
1.
2.
3.
4.
S inside plane and P inside plane
Save p in the output-List
S inside plane and P outside plane
Find intersection point i
Save i in the output-List
S outside plane and P outside plane
Save nothing
S outside plane and P inside plane
Find intersection point i
Save i in the output-List, followed by P
ALGORITHHM
1.
2.
3.
4.
Read polygon vertices
Read window coordinates
For every edge of window do
Check every edge of the polygon to do 4 tests
1.
2.
5.
6.
Save the resultant vertices and the intersections
in the output list.
The resultant set of vertices is then sent for
checking against next boundary.
Draw polygon using output-list.
Stop.
GENERALIZED CLIPPING
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We have used four clipping routines; one for each
boundary. But these routines are identical. They
differ only in their tests for determining whether
a point is inside or outside the boundary.
Can we make it general ??
It is possible to write one single routine and pass
information about boundary as parameter. This
change allows to clip along any line(not just
horizontal or vertical boundaries).
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Such general routine can be then used to clip
along an arbitrary convex polygon.
Fig :A window with 6 clipping boundaries
VIEWING TRANSFORMATION
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An image is defined in a world coordinate system,
WCS i.e. Using Cartesian coordinate system.
World Coordinate system: application-specific
Example: drawing dimensions may be in meters,
km, feet, etc.
 When picture is represented on display device,it
is measured in physical device coordinate
system(PDCS) corresponding to the display
device.
 Viewing
Transformation:
Mapping
of
coordinates is acheived with the use of coordinate
transformation
i.e. Maps picture coordinates in the WCS to display
coordinates in PDCS.
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NORMALIZED COORDINATES
o
o
o
o
o
o
Different display devices have different screen sizes and
resolutions, which is measured in pixels.
Picture on a screen with high resolution appears small in
size, where as same pic on screen with less resolution
appears big in size.
To avoid this we make our program device independent.
So picture coordinates have to be represented in some
units other than pixels.
Device independent units are called as Normalized
device coordinates.
In this screen measures 1 unit wide and 1 unit length.
TRANSFORMATION TO NORMALIZED COORDINATES
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Interpreter is used to convert normalized co-ordinates into device coordinates. A simple linear formula is used for this.
x = xn * Xw
y = yn * Yw
Where : x= actual device x co-or
y= actual device y co-or
xn = Normalized x co-or
yn = Normalized y co-or
Xw =Width of actual screen in Pixels
Yw = Height of actual screen in Pixels
Workstation Transformation:
Transformation of object description from normalized coordinates to
device coordinates.
It is accomplished by selecting a window area in normalized space
and viewport area in coordinates of display device.
 WCS is infinite in extent and device display area is finite.
 A world coordinate area selected for display is called a window.
( defines what to display)
 An area on a device to which a window is mapped is called
viewport. ( defines where to display.)
ywmax
Clipping
Window
Viewport
yvmax
yvmin
ywmin
xwmin
xwmax
xvmin
xvmax
WINDOW TO VIEWPORT COORDINATE TRANSFORMATION
Would like to:
Specify drawing in world coordinates
Display in screen coordinates
Need some sort of mapping
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Called Window-to-viewport mapping
Basic W-to-V mapping steps:
Define a world window
Define a viewport
Compute a mapping from window to viewport
WINDOW TO VIEWPORT COORDINATE TRANSFORMATION
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Fig shows mapping of object from window to viewport.
A point at (xw , yw ) in window is mapped into position (xv , yv ) in
the associative viewport.
WINDOW TO VIEWPORT COORDINATE TRANSFORMATION
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To maintain relative replacement in viewport as in window we
require,
xv  xvmin
xw  xwmin

xvmax  xvmin xwmax  xwmin
yv  yvmin
yw  ywmin

yvmax  yvmin ywmax  ywmin
Solving these equations for viewport position
xv = xvmin + ( xw – xwmin ) sx
yv = yvmin + ( yw – ywmin ) sy
Where
Scaling factors:
xvmax  xvmin
sx 
,
xwmax  xwmin
yvmax  yvmin
sy 
ywmax  ywmin
WINDOW TO VIEWPORT COORDINATE TRANSFORMATION
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1.
2.
Mapping of window to viewport require following
transformations
Scaling using fixed point position of (xwmin , ywmin ) that scales
the window area to the size of viewport area.
Translation : translating scaled window area to the position of
viewport area.
WINDOW TO VIEWPORT COORDINATE TRANSFORMATION
3D VIEWING TRANSFORMATION