Cs8092 computer graphics and multimedia unit 2

UNIT II
TWO-
DIMENSIONAL
GRAPHICS

CONTENT
Two dimensional geometric transformations — Matrix
representations and homogeneous coordinates, composite
transformations; Two dimensional viewing — viewing pipeline,
viewing coordinate reference frame; window-to-viewport
coordinate transformation, Two dimensional viewing functions;
clipping operations — point, line, and polygon clipping
algorithms.

Transformation
Transform every point on an object according to certain rule.
Initial Object
Transformed Object
P (x,y)
Q (x’, y’)
x x’
y y’
T
The point Q is the image of P under the
transformation T.

Translation
  
  
x x t
y y t
x
y
(10,5) (30,5)
(20,35)
(55,60)
(65,30)
(45,30)
The vector (tx, ty) is called the offset vector.

Translation (OpenGL)
Specifying a 2D-Translation:
glTranslatef(tx, ty, 0.0);
(The z component is set to 0 for 2D translation).

(x,y)
(x’,y’)
q
  
  
x x y
y x y
cos sin
sin cos
q q
q q
o
(x,y)
(x’,y’)
x
y
Rotation About the Origin
The above 2D rotation is actually a rotation about the
z-axis (0,0,1) by an angle q.

Rotation About the Origin
Specifying a 2D-Rotation about the origin:
glRotatef(theta, 0.0, 0.0, 1.0);
theta: Angle of rotation in degrees.
The above function defines a rotation about the z-axis (0,0,1).

(xp , yp)
(x,y)
(x’,y’)
Pivot Point
•Pivot point is the point of rotation
•Pivot point need not necessarily be on the object
Rotation About a Pivot Point

(xp , yp)
p
p
y
y
y
x
x
x




1
1
(x,y)
(x1, y1)
STEP-1: Translate the pivot point to the origin

q
q
q
q
cos
1
sin
1
2
sin
1
cos
1
2
y
x
y
y
x
x




(x1, y1)
STEP-2: Rotate about the origin
(x2, y2)

p
p
y
y
y
x
x
x






2
2
STEP-3: Translate the pivot point to original position
(x2, y2)
(xp, yp)
(x’, y’)

     
     
x x x y y x
y x x y y y
p p p
p p p
( )cos ( )sin
( )sin ( )cos
q q
q q
Specifying a 2D-Rotation about a pivot point (xp,yp):
glTranslatef(xp, yp, 0);
glRotatef(theta, 0, 0, 1.0);
glTranslatef(-xp, -yp, 0);
Note the OpenGL specification of the sequence of transformations in the
reverse order !

 
 
x x s
y y s
x
y
.
.
s s
x y

Uniform Non-Uniform
s s
x y

( , )
s s
x y  0
(x,y)
(x’,y’)
(x,y)
(x’,y’)
Scaling About the Origin
The parameters sx, sy are called scale factors.

Specifying a 2D-Scaling with respect to the origin:
glScalef(sx, sy, 1.0);
sx, sy: Scale factors along x, y.
For proper scaling sx, sy must be positive.
For 2D scaling, the third scale factor must be set to 1.0.
Scaling About the Origin

(xf , yf )
f
y
f
f
x
f
y
s
y
y
y
x
s
x
x
x








).
(
).
(
(x,y)
(x’,y’)
• Translate the fixed point to origin
• Scale with respect to the origin
•Translate the fixed point to its original
position.
Scaling About a Fixed Point

Reflections Initial
Object
Reflection about x
y =  y
x
y
Reflection about
origin
x =  x
y =  y
Reflection about y
x =  x

Reflections
Reflection about x: glScalef(1, -1, 1);
Reflection about y: glScalef(-1, 1, 1);
Reflection about origin: glScalef(-1, -1, 1);

Shear
  
 
x x h y
y y
x
•A shear transformation in the x-direction (along x)
shifts the points in the x-direction proportional
to the y-coordinate.
•The y-coordinate of each point is unaffected.








 





 













 




















 












x
y
x
y
t
t
x
y
x
y
x
y
s
s
x
y
x
y
x
y
cos sin
sin cos
q q
q q
0
0
Translation
Rotation [Origin]
Scaling [Origin]
Matrix Representations



































































y
x
y
x
y
x
y
x
y
x
y
x
1
0
0
1
1
0
0
1
1
0
0
1
Reflection about x
Reflection about y
Reflection about
the Origin











































y
x
h
y
x
y
x
h
y
x
1
0
1
1
0
1
Shear along x
Shear along y

Homogeneous Coordinates
To obtain square matrices an additional row was added to the matrix and an
additional coordinate, the w-coordinate, was added to the vector for a point. In
this way a point in 2D space is expressed in three-dimensional homogeneous
coordinates.
This technique of representing a point in a space whose dimension is one
greater than that of the point is called homogeneous representation. It
provides a consistent, uniform way of handling affine transformations.

Cartesian Homogeneous
0
),
,
,
(
,
0
),
,
,
(
)
,
(












c
c
b
a
c
b
c
a
h
h
yh
xh
y
x
Examples: (5, 8) (15, 24, 3)
(x, y) (x, y, 1)
•If we use homogeneous coordinates, the geometric
transformations given above can be represented using
only a matrix pre-multiplication.
• A composite transformation can then be represented
by a product of the corresponding matrices.














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

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
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





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
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












































x
y
t
t
x
y
x
y
x
y
x
y
s
s
x
y
x
y
x
y
1
1 0
0 1
0 0 1 1
1
0
0
0 0 1 1
1
0 0
0 0
0 0 1 1
cos sin
sin cos
q q
q q
Translation
P’=TP
Rotation [O]
P’=RP
Scaling [O]
P’=SP
Basic Transformations
























1
]
[
1
y
x
T
y
x
If,
























1
]
[
1
1
y
x
T
y
x
then,
)
(
)
(
1
,
1
)
,
(
)
(
)
(
)
,
(
)
,
(
1
1
1
1
h
H
h
H
s
s
S
s
s
S
R
R
t
t
T
t
t
T
x
x
y
x
y
x
y
x
y
x




















q
q
Examples:
Inverse of Transformations

)
,
(
)
,
(
)
,
(
1
)
(
)
(
)
(
)
(
)
,
(
)
,
(
)
,
(
2
1
2
1
y
y
x
x
y
x
y
x
y
y
x
x
y
x
y
x
w
s
w
s
S
w
w
S
s
s
S
R
R
R
R
u
t
u
t
T
u
u
T
t
t
T







q
q
q
q
q
Additional Properties:
Transformation Matrices

Transformation T followed by
Transformation Q followed by
Transformation R: 






















1
]
][
][
[
1
y
x
T
Q
R
y
x
Example: (Scaling with respect to a fixed point)























































x
y
x
y
s
s
x
y
x
y
f
f
x
y
f
f
1
1 0
0 1
0 0 1
0 0
0 0
0 0 1
1 0
0 1
0 0 1 1
Composite Transformations
Order of Transformations

In composite transformations, the order of
transformations is very important.
Order of Transformations
Rotation followed by Translation:
Translation followed by Rotation:

OpenGL postmultiplies the current matrix
with the new transformation matrix
Order of Transformations (OpenGL)
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glTranslatef(tx, ty, 0);
glRotatef(theta, 0, 0, 1.0);
glVertex2f(x,y);
Rotation followed by Translation !!
Current Matrix
[ I ]
[ T ]
[ T ] [ R ]
[ T ] [ R ] P

Line Angle Distance Area
Translation Yes Yes Yes Yes
Rotation Yes Yes Yes Yes
Scaling Yes No No No
Reflection Yes Yes Yes Yes
Shear Yes No No Yes
Preserved Attributes
General Properties

A general invertible, linear, transformation.
   
   
 
x a x b y c
y a x b y c
a b a b
1 1 1
2 2 2
1 2 2 1 0
( )
Affine Transformation
Transformation Matrix:










1
0
0
2
2
2
1
1
1
c
b
a
c
b
a

Concatenation.
We perform 2 translations on the same point:
)
,
(
)
,
(
)
,
(
:
expect
we
So
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
2
1
2
1
2
2
1
1
2
1
2
1
2
2
1
1
2
2
1
1
y
y
x
x
y
x
y
x
y
y
x
x
y
x
y
x
y
x
y
x
d
d
d
d
T
d
d
T
d
d
T
P
d
d
d
d
T
P
d
d
T
d
d
T
P
P
d
d
T
P
P
d
d
T
P






















Concatenation.
?
1
0
0
1
0
0
1
.
1
0
0
1
0
0
1
:
is
)
,
(
)
,
(
product
matrix
The
2
2
1
1
2
2
1
1






















y
x
y
x
y
x
y
x
d
d
d
d
d
d
T
d
d
T
Matrix product is variously referred to as compounding,
concatenation, or composition

Concatenation.
Matrix product is variously referred to as compounding, concatenation, or
composition.


































1
0
0
1
0
0
1
1
0
0
1
0
0
1
.
1
0
0
1
0
0
1
:
is
)
,
(
)
,
(
product
matrix
The
2
1
2
1
2
2
1
1
2
2
1
1
y
y
x
x
y
x
y
x
y
x
y
x
d
d
d
d
d
d
d
d
d
d
T
d
d
T

Properties of translations.
)
,
(
)
,
(
T
4.
)
,
(
)
,
(
)
,
(
)
,
(
3.
)
,
(
)
,
(
)
,
(
2.
)
0
,
0
(
1.
1
-
y
x
y
x
y
x
y
x
y
x
y
x
y
y
x
x
y
x
y
x
s
s
T
s
s
s
s
T
t
t
T
t
t
T
s
s
T
t
s
t
s
T
t
t
T
s
s
T
I
T











Note : 3. translation matrices are commutative.

Homogeneous form of scale.







y
x
y
x
s
s
s
s
S
0
0
)
,
(
Recall the (x,y) form of Scale :











1
0
0
0
0
0
0
)
,
( y
x
y
x s
s
s
s
S
In homogeneous coordinates :

Two-Dimensional Viewing
37
Co-ordinate Systems.
 Cartesian – offsets along the x and y axis from (0.0)
 Polar – rotation around the angle q.
 Graphic libraries mostly using Cartesian co-ordinates
 Any polar co-ordinates must be converted to Cartesian co-
ordinates
 Four Cartesian co-ordinates systems in computer Graphics.
1. Modeling co-ordinates
 2. World co-ordinates
 3. Normalized device co-ordinates
 4. Device co-ordinates

Modeling Coordinates
38
 Also known as local coordinate.
 Ex: where individual object in a scene within separate
coordinate reference frames.
 Each object has an origin (0,0)
 So the part of the objects are placed with reference to the
object’s origin.
 In term of scale it is user defined, so, coordinate values can be
any size.

World Co-ordinates.
39
 The world coordinate system describes the relative positions
and orientations of every generated objects.
 The scene has an origin (0,0).
 The object in the scene are placed with reference to the
scenes origin.
 World co-ordinate scale may be the same as the modeling co-
ordinate scale or it may be different.
 However, the coordinates values can be any size (similar to
MC)

Normalized Device Co-ordinates
40
 Output devices have their own co-ordinates.
 Co-ordinates values:
 The x and y axis range from 0 to 1
 All the x and y co-ordinates are floating point numbers in the
range of 0 to 1
 This makes the system independent of the various devices
coordinates.
 This is handled internally by graphic system without user
awareness.

Device Co-ordinates
41
 Specific co-ordinates used by a device.
 Pixels on a monitor
 Points on a laser printer.
 mm on a plotter.
 The transformation based on the individual device is handled
by computer system without user concern.

42
Example:
 Graphic program which draw an entire building by an
architect but we only interested on the ground floor
 Map of sales for entire region but we only like to
know from certain region of the country.

43
 When we interested to display certain portion of the drawing,
enlarge the portion, windowing technique is used
 Technique for not showing the part of the drawing which one is
not interested is called clipping
 An area on the device (ex. Screen) onto which the window will
be mapped is called viewport.
 Window defines what to be displayed.
 A viewport defines where it is to be displayed.
 Most of the time, windows and viewports are usually rectangles
in standard position(i.e aligned with the x and y axes). In some
application, others such as general polygon shape and circles
are also available
 However, other than rectangle will take longer time to process.

Viewing Transformation
44
 Viewing transformation is the mapping of a part of a world-
coordinate scene to device coordinates.
 In 2D (two dimensional) viewing transformation is simply
referred as the window-to-viewport transformation or the
windowing transformation.
 Mapping a window onto a viewport involves converting from
one coordinate system to another.
 If the window and viewport are in standard position, this just
 involves translation and scaling.
 if the window and/or viewport are not in standard, then
extra transformation which is rotation is required.

Viewing Transformation
45
0
1
1
x-world
y-world
window window
Normalised device
world
y-view
x-view

Window-To-Viewport Coordinate Transformation
46
Window-to-Viewport transformation

47
.
XWmax
YWmax
XWmin
YWmin
XVmax
XVmin
YVmax
YVmin
xw,yw
xv,yv

48
xv - xvmin = xw - xwmin
xvmax - xvmin xwmax - xwmin
yv – yvmin = yw - ywmin
yvmax – yvmin ywmax - ywmin
From these two equations we derived
xv = xvmin + (xw – xwmin)sx
yv = yvmin + (yw – ywmin)sy
where the scaling factors are
sx = xvmax – xvmin sy = yvmax - yvmin
xwmax – xwmin ywmax - ywmin

49
The sequence of transformations are:
1. Perform a scaling transformation using a
fixed-point position of (xwmin,ywmin) that
scales the window area to the size of the
viewport.
2. Translate the scaled window area to the
position of the viewport.

50
 Relative proportions of objects are maintained if the scaling
factors are the same (sx = sy). Otherwise, world objects will be
stretched or contracted in either x or y direction when displayed
on output device.
 How about character strings when map to viewport?
 maintains a constant character size (apply when
standard character fonts cannot be changed).
 If character size can be changed, then windowed will be
applied like other primitives.
 For characters formed with line segments, the mapping
to viewport is carried through sequence of line
transformations .

Viewport-to-Normalized Device Coordinate
Transformation
51
 From normalized coordinates, object descriptions can
be mapped to the various display devices
 When mapping window-to-viewport transformation is
done to different devices from one normalized space, it is
called workstation transformation.

OpenGL 2D Viewing Functions
• To transform from world coordinate to screen coordinates, the appropriate
matrix mode must be chosen
• glMatrixMode (GL_PROJECTION);
• glLoadIdentity( );
• To define a 2D clipping window, we use OpenGL Utility function:
• gluOrtho2D( xwmin, xwmax, ywmin, ywmax);
• This function also perform normalization (NDC)
53

OpenGL 2D Viewing Functions
• To specify the viewport parameters in OpenGL, we use function
• glViewport(xvmin, yvmin, vpWidth, vpHeight);
54

2D Clipping
1. Introduction
2. Point Clipping
3. Line Clipping
4. Polygon/Area Clipping
5. Text Clipping
6. Curve Clipping

2D Clipping
1. Introduction:
A scene is made up of a collection of objects specified in world
coordinates
World Coordinates

2D Clipping
When we display a scene only those objects within a particular
window are displayed
wymax
wymin
wxmin wxmax
Window
World Coordinates

2D Clipping
Because drawing things to a display takes time we clip
everything outside the window
wymax
wymin
wxmin wxmax
World Coordinates
Window

1.1 Definition:
– Clipping is the process of determining which elements of
the picture lie inside the window and are visible.
– By default, the “clip window” 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 window” to
a smaller portion of the canvas
• partial canvas redraw for menus, dialog boxes, other obscuration
2D Clipping

1.2 Shielding:
– Shielding or exterior clipping is the reverse operation of
clipping where window act as the block used to abstract the
view.
– Examples
• A multi view window system
• The design of page layouts in advertising or publishing
applications or for adding labels or design patterns to
picture.
• Combining graphs, maps o schematics
2D Clipping

2D Clipping
wymax
wymin
wxmin wxmax
Window
World Coordinates

2D Clipping
1.3 Example:
For the image below consider which lines and points should be kept
and which ones should be clipped against the clipping window
wymax
wymin
wxmin wxmax
Window
P1
P2
P3
P6
P5
P7
P10
P9
P4
P8

1.4 Applications:
– Extract part of a defined scene for viewing.
– Drawing operations such as erase, copy, move etc.
– Displaying multi view windows.
– Creating objects using solid modeling techniques.
– Anti-aliasing line segments or object boundaries.
– Identify visible surfaces in 3D views.
2D Clipping

2D Clipping
1.5 Types of clipping:
– Three types of clipping techniques are used depending
upon when the clipping operation is performed
a. Analytical clipping
– Clip it before you scan convert it
– used mostly for lines, rectangles, and polygons, where
clipping algorithms are simple and efficient

2D Clipping
b. Scissoring
– Clip it during scan conversion
– 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
• if primitive is not much larger than clip region, most pixels will fall
inside
• can be more efficient than analytical clipping.

2D Clipping
c. Raster Clipping
– Clip it after scan conversion
– render everything onto a temporary canvas and copy the
clipping region
• wasteful, but simple and easy,
• often used for text

2D Clipping
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

1.6 Levels of clipping:
– Point Clipping
– Line Clipping
– Polygon Clipping
– Area Clipping
– Text Clipping
– Curve Clipping
2D Clipping

Point Clipping
– Simple and Easy
– a point (x,y) is not clipped if:
wxmin ≤ x ≤ wxmax
&
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

Line Clipping
– It is Harder than point
clipping
– We first examine the end-
points of each line to see
if they are in the window
or not
• Both endpoints inside, line
trivially accepted
• One in and one out, line is
partially inside
• Both outside, might be
partially inside
• What about trivial cases?
ymin
ymax
xmin xmax

Line Clipping
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!

2D Line Clipping Algorithms
1. Analytical Line Clipping
2. Cohen Sutherland Line Clipping
3. Liang Barsky Line Clipping

Cohen-Sutherland Line Clipping
– An efficient line clipping
algorithm
– The key advantage of the
algorithm is that it vastly
reduces the number of line
intersections that must be
calculated.
Dr. Ivan E. Sutherland co-
developed the Cohen-
S u t h e r l a n d c l i p p i n g
algorithm. Sutherland is a
graphics giant and includes
amongst his achievements
the invention of the head
m o u n t e d d i s p l a y .
Cohen is something of a mystery – can
a n y b o d y f i n d o u t w h o h e w a s?

– Two phases Algorithm
Phase I: Identification Phase
All line segments fall into one of the following categories
1. Visible: Both endpoints lies inside
2. Invisible: Line completely lies outside
3. Clipping Candidate: A line neither in category 1 or 2
Phase II: Perform Clipping
Compute intersection for all lines that are candidate for
clipping.

Phase I: Identification Phase: World space is divided into regions
based on the window boundaries
– Each region has a unique four bit region code
– Region codes indicate the position of the regions with respect to the
window
1001 1000 1010
0001
0000
Window
0010
0101 0100 0110
above below right left
3 2 1 0
Region Code Legend

Every end-point is labelled with the appropriate region code
wymax
wymin
wxmin wxmax
Window
P3 [0001]
P6 [0000]
P5 [0000]
P7 [0001]
P10 [0100]
P9 [0000]
P4 [1000]
P8 [0010]
P12 [0010]
P11 [1010]
P13 [0101] P14 [0110]

Visible Lines: Lines completely contained within the window
boundaries have region code [0000] for both end-points so are
not clipped
wymax
wymin
wxmin wxmax
Window
P3 [0001]
P6 [0000]
P5 [0000]
P7 [0001]
P10 [0100]
P9 [0000]
P4 [1000]
P8 [0010]
P12 [0010]
P11 [1010]
P13 [0101] P14 [0110]

Invisible Lines: Any line with a common set bit in the region
codes of both end-points can be clipped completely
– The AND operation can efficiently check this
– Non Zero means Invisible
wymax
wymin
wxmin wxmax
Window
P3 [0001]
P6 [0000]
P5 [0000]
P7 [0001]
P10 [0100]
P9 [0000]
P4 [1000]
P8 [0010]
P12 [0010]
P11 [1010]
P13 [0101] P14 [0110]

Clipping Candidates: Lines that cannot be identified as
completely inside or outside the window may or may not cross
the window interior. These lines are processed in Phase II.
– If AND operation result in 0 the line is candidate for clipping
wymax
wymin
wxmin wxmax
Window
P3 [0001]
P6 [0000]
P5 [0000]
P7 [0001]
P10 [0100]
P9 [0000]
P4 [1000]
P8 [0010]
P12 [0010]
P11 [1010]
P13 [0101] P14 [0110]

Assigning Codes
– Let point (x,y) is be given
code b3b2b1b0:
bit 3 = 1 if wymax - y ≤0
bit 2 = 1 if y - wymin ≤ 0
bit 1 = 1 if wxmax - x ≤0
bit 0 = 1 if x - wxmin ≤ 0
wymax
wymin
wxmin wxmax
Window
P3 [0001]
P6 [0000]
P5 [0000]
P7 [0001]
P10 [0100]
P9 [0000]
P4 [1000]
P8 [0010]
P12 [0010]
P11 [1010]
P13 [0101] P14 [0110]

Cohen-Sutherland Clipping Algorithm
Phase II: Clipping Phase: Lines that are in category 3 are now
processed as follows:
– Compare an end-point outside the window to a boundary
(choose any order in which to consider boundaries e.g. left,
right, bottom, top) and determine how much can be
discarded
– If the remainder of the line is entirely inside or outside the
window, retain it or clip it respectively
– Otherwise, compare the remainder of the line against the
other window boundaries
– Continue until the line is either discarded or a segment
inside the window is found

• Intersection points with the window boundaries are calculated
using the line-equation parameters
– Consider a line with the end-points (x1, y1) and (x2, y2)
– The y-coordinate of an intersection with a vertical window
boundary can be calculated using:
y = y1 + m (xboundary - x1)
where xboundary can be set to either wxmin or wxmax
– The x-coordinate of an intersection with a horizontal
window boundary can be calculated using:
x = x1 + (yboundary - y1) / m
where yboundary can be set to either wymin or wymax

• We can use the region codes to determine which window
boundaries should be considered for intersection
– To check if a line crosses a particular boundary we
compare the appropriate bits in the region codes of its end-
points
– If one of these is a 1 and the other is a 0 then the line
crosses the boundary.

Example1: Consider the line P9 to P10 below
– Start at P10
– From the region codes
of the two end-points we
know the line doesn’t
cross the left or right
boundary
– Calculate the intersection
of the line with the bottom boundary
to generate point P10’
– The line P9 to P10’ is completely inside the window so is
retained
wymax
wymin
wxmin wxmax
Window
P10 [0100]
P9 [0000]
P10’ [0000]
P9 [0000]

Example 2: Consider the line P3 to P4 below
– Start at P4
– From the region codes
of the two end-points
we know the line
crosses the left
boundary so calculate
the intersection point to
generate P4’
– The line P3 to P4’ is completely outside the window so is
clipped
wymax
wymin
wxmin wxmax
Window
P4’ [1001]
P3 [0001]
P4 [1000]
P3 [0001]

Example 3: Consider the line P7 to P8 below
– Start at P7
– From the two region
codes of the two
end-points we know
the line crosses the
left boundary so
calculate the
intersection point to
generate P7’
wymax
wymin
wxmin wxmax
Window
P7’ [0000]
P7 [0001] P8 [0010]
P8’ [0000]

Example 4: Consider the line P7’ to P8
– Start at P8
– Calculate the
intersection with the
right boundary to
generate P8’
– P7’ to P8’ is inside
the window so is
retained
wymax
wymin
wxmin wxmax
Window
P7’ [0000]
P7 [0001] P8 [0010]
P8’ [0000]

Mid-Point Subdivision Method
– Algorithm
1. Initialise the list of lines to all lines
2. Classify lines as in Phase I
i. Assign 4 point bit codes to both end points a3a2a1a0 and b3b2b1b0
ii. If (a3a2a1a0 = b3b2b1b0 = 0 )Line in category 1
iii. If (a3a2a1a0)AND (b3b2b1b0 ) # 0 ) Line in category 2
iv. If (a3a2a1a0)AND (b3b2b1b0 ) = 0 ) Line in category 3
3. Display all lines from the list in category 1 and remove;
4. Delete all lines from the list in category 2 as they are invisible;
5. Divide all lines of category 3 are into two smaller segments at mid-point
(xm,ym) where xm = (x1 +x2)/2 and ym = (y1 +y2)/2
6. Remove the original line from list and enter its two newly created
segments.
7. Repeat step 2-5 until list is null.

wymax
wymin
wxmin wxmax
Window

Mid-Point Subdivision Method
– Integer Version
– Fast as Division by 2 can be performed by simple shift
right operation
– For NxN max dimension of line number of subdivisions
required log2 N.
– Thus a 1024x1024 raster display require just 10
subdivisions………

2D Clipping
1. Introduction
2. Point Clipping
3. Line Clipping
4. Polygon / Area Clipping
5. Text Clipping
6. Curve Clipping

Polygon Clipping
• Note the difference between clipping lines and
polygons:
NOTE!

Polygon Clipping
• Some difficulties:
– Maintaining correct inside/outside
– Variable number of vertices
– Handle screen corners
correctly

Sutherland-Hodgman Area Clipping
• 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

1. Basic Concept:
• Simplify via separation
• Clip whole polygon against one edge
– Repeat with output for other 3 edges
– Similar for 3D
• You can create intermediate vertices that get thrown
out
Sutherland-Hodgeman Polygon Clipping

• Example
Start Left Right Bottom Top
Note that the point one of the points added when clipping
on the right gets removed when we clip with bottom

2. Algorithm:
Let (P1, P2,…. PN) be the vertex list of the Polygon to be
clipped and E be the edge of +vely oriented, convex clipping
window.
We clip each edge of the polygon in turn against each window
edge E, forming a new polygon whose vertices are determined
as follows:

Four cases
1. Inside: If both Pi-1 and Pi are to the left of window edge
vertex then Pi is placed on the output vertex list.
2. Entering: If Pi-1 is to the right of window edge and Pi is to
the left of window edge vertex then intersection (I) of Pi-1 Pi
with edge E and Pi are placed on the output vertex list.
3. Leaving: If Pi-1 is to the left of window edge and Pi is to the
right of window edge vertex then only intersection (I) of Pi-1
Pi with edge E is placed on the output vertex list.
4. Outside: If both Pi-1 and Pi are to the right of window edge
nothing is placed on the output vertex list.

Creating New Vertex List
out  out
save nothing
Outside
(0 output)
Pi-1
Pi
in  in
save ending vert
Inside
(1 output)
Pi-1
Pi
out  in
save new clip vert
and ending vert
Entering
(2 outputs)
Pi
Pi-1
Pi
in  out
save new clip vert
Leaving
(1 output)
Pi-1

Sutherland-Hodgman Polygon Clipping
• 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

START
INPUT VERTEX LIST
(P1, P2........, PN)
IF PiPi+1
INTERSECT E ?
FOR i =1 TO (N-1) DO
COMPUTE I
OUTPUT I IN VERTEX LIST
IF Pi TO LEFT
OF E ?
YES NO
YES
OUTPUT Pi IN VERTEX LIST
NO
i = i+1
Flow Chart
Special case for
first Vertex

Flow Chart
Special case for
first Vertex
IF PNP0
INTERSECT
E ?
COMPUTE I
OUTPUT I IN VERTEX
LIST
YES NO
END
YOU CAN ALSO APPEND AN ADDITIONAL VERTEX
PN+1 = P1 AND AVOID SPECIAL CASE FOR FIRST
VERTEX

Inside/Outside Test:
Let P(x,y) be the polygon vertex which is to be tested against
edge E defined form A(x1, y1) to B(x2, y2). Point P is to be said
to the left (inside) of E or AB iff
or C = (x2 – x1) (y – y1) – (y2 – y1)(x – x1) > 0
otherwise it is said to be the right/Outside of edge E
0
1
2
1
1
2
1






x
x
x
x
y
y
y
y

Weiler-Atherton Polygon Clipping
• Problem with Sutherland-Hodgeman:
– Concavities can end up linked
• Weiler-Atherton creates separate polygons in such
cases
Remember
me?

• Example
add clip pt.
and end pt.
add end pt. add clip pt.
cache old dir.
follow clip edge until
a) new crossing
found
b) reach pt. already
added

• Example (cont)
continue from
cached location
add clip pt.
and end pt.
add clip pt.
cache dir.
follow clip edge until
a) new crossing
found
b) reach pt. already
added

• Example (concluded)
continue from
cached location
nothing added
finished
Final result:
Two unconnected
polygons

• Difficulties:
– What if the polygon re-crosses edge?
– How many “cached” crosses?
– Your geometry step must be able to create new polygons
• Instead of 1-in-1-out

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 Window
Window Window

Text Clipping
Text clipping relies on the concept of bounding
rectangle
TYPES
1. All or None String Clipping
2. All or None Character Clipping
3. Component Character Clipping

Text Clipping
1. All or None String Clipping
• In this scheme, if all of the string is inside window, we clip it,
otherwise the string is discarded. This is the fastest method.
• The procedure is implemented by consider a bounding rectangle
around the text pattern. The boundary positions are compared to
the window boundaries. In case of overlapping the string is
rejected.
STRI NG 1
STRING 3
STRING 4
STRING 5
STRING 4

Text Clipping
2. All or None Character Clipping
• In this scheme, we discard only those characters that are not
completely inside window.
• Boundary limits of individual characters are compared against
window. In case of overlapping the character is rejected.
STRI NG 1
STRING 3
STRING 4
STRING 5
NG 1
TRING 3
STRING 4

Text Clipping
3. Component Character Clipping
• Characters are treated like graphic objects.
– Bit Mapped Fonts : Point Clipping
– Outlined Fonts : Line/Curve Clipping
• In case of overlapping the part of the character inside is displayed
and the outside portion of the character is rejected.
STRI NG 1
STRING 3
STRING 4
STRING 5
NG 1
STRING 4
STRING 3

Curve Clipping
– Areas with curved boundaries can be clipped with
methods similar to line and polygon clipping.
– Curve clipping requires more processing as it
involve non linear equations.
– Bounding Rectangles are used to test for overlap
with rectangular clip window.

– If bounding rectangle is completely inside the
object/curve is saved.
– If bounding rectangle is completely outside the
object/curve is discarded.
Curve Clipping

– If both the above tests fails we use other
computation saving approaches depending upon
type of object
• Circle: Use coordinate extent of individual quadrant,
then octant if required.
• Ellipse: Use coordinate extent of individual quadrant.
• Point: Use point clipping
Curve Clipping

Cs8092 computer graphics and multimedia unit 2

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