3.ComputerGraphics

CAD/CAM Principles and Applications by P N Rao, 2nd Ed

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CAD/CAM Principles and Applications

Ch 3 Computer Graphics


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Objectives• Convert vector straight lines to raster images to be displayed on

a raster terminal utilizing the pixel information• Understand the problems associated with displaying vectorial

information on a raster terminal• Various types of coordinate systems used in displaying CAD

information• The data requirements of a graphic image and the database

storage methods used • Different types of geometric transformations used during CAD

geometry generation and display and their evaluation• Mathematics required to display a 3D image on the 2D screen of

the display device• Understand the problems associated with the display of graphic

images in the display screen such as clipping and hidden line elimination


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3.1 Raster Scan Graphics

• DDA or Digital Differential Analyser is one of the first algorithms developed for rasterising the vectorial information. The equation of a straight line is given by

• Y = m X + C


4

Fig. 3.1 A straight line drawing

X

Y

x1x2

y1

y2


5

Fig. 3.2 Flow chart for line drawing calculation procedure

Calculate dx = x2 - x1dy = y2 - y1

If |dx| > |dy| St = |dx|

dx = dx / STdy = dy / ST

X = xiY = yi

Set pixel at X, Y

End of Loop?

STOP

Yes

ST = |dy|

No

X = X + dxY = Y + dy

Yes

No


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3.1.2 Bresenham's Algorithm

• Bresenham's method is an improvement over DDA since it completely eliminates the floating-point arithmetic except for the initial computations.

• All other computations are fully integer arithmetic thus is more efficient for raster conversion


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Fig. 3-3 Line drawing using Bresenhamalgorithm

X

Y

i i+1

y d2

d1i

i+1


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Fig. 3.4 Flow chart for line drawing calculation using Bresenham

procedure

Calculate dx = x2 - x1dy = y2 - y1C1 = 2 dy

C2 = 2 (dy - dx)

x = x + 1

STOP

Put a pixel at (x1, y1)

P(i+1) = P(i) + C1y(i+1) = y(i)

x = x1y = y1

p1 = 2 dy - dx

If Pi < 0 P(i+1) = Pi + C2Y(i+1) = y(i) + 1No

Yes

No End of loop?

Yes


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3.1.3 Antialiasing lines


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Fig. 3.5 The staircase effect of pixels when

drawing inclined lines


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Fig. 3.6 The staircase effect of pixels when drawing

inclined lines decreases with increased resolution


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Fig. 3.7 Antialiasing of pixels proportional to

the portion of pixel occupied by the line

1

6

7

8

1

234

5

2 3 4 65 7 8


13

Fig. 3.8 Unequal number of lines displayed

with the same number of pixels


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3.2 Co-ordinate systems


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Fig. 3.9 A typical component to be modelled

150

120

9050

60

2040

30

40


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World Co-ordinate System

• This refers to the actual co-ordinate system used as master for the component.

• Some times it may also be called as model co-ordinate system.


17

Fig. 3.10 A typical component with its

associated WCS

X

Y

Z


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User Co-ordinate System

• However, sometimes it becomes difficult to define certain geometries if they are to be defined from the WCS. In such cases alternate co-ordinate systems can be defined relative to the WCS. These co-ordinate systems are termed as user co-ordinate systems (UCS) or working co-ordinate systems.


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Fig. 3.11 A typical component with its

associated UCS

X

Y

Z

X'

Y'

Z'


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Display Co-ordinates

• This refers to the actual co-ordinates to be used for displaying the image on the screen.


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Fig. 3.12 A typical component with its various

view positions

X

Y

Z

FRONT

TOP

RIGHT SIDE


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Fig. 3.13 Various views generated from the

model shown in Fig 3.12

150

120

90

50 60

30

60

20

20

40

TOP

FRONT RIGHT SIDE

XY

Z

X

Y

Z

XY

Z


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3.3 Database Structures for Graphic Modelling


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3.3 Database Structures for Graphic Modelling

• Organisational data• Identification number,• Drawing number,• Design origin and status

of changes,• Current status,• Designer name,• Date of design,• Scale,• Type of projections,• Company.

• Technological data• Geometry,• Dimensions,• Tolerances,• Surface finishes,• Material specifications

or reference,• Manufacturing

procedures,• Inspection procedures.


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Fig. 3.14 Data structure for geometric modelsSOLID

EDGE

VERTEX

SURFACE

VERTEX SURFACE


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Fig. 3.15 Complete data structure for

geometric models of productsProduct

Assembly

Single parts

Solid body

Faces

Edges

Vertices

Dimensions Attributes


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Fig. 3.16 Relational data structure for

geometric models

solid body Face list Edges Vertices

X Y Z


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3.4 Transformation of Geometry

• Translation• Scaling• Reflection or Mirror• Rotation


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Fig. 3-17 Some of the possible geometric

transformations

X

Y

Y

X

Y

X

Y

X

dX

dYTranslation Scaling

ReflectionRotation

30°

2525


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Fig. 3.18 Translation of geometry

X

Y

Z

X'

Y'

Z'

P

P*

X

Y P


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Fig. 3.19 Translation of geometry in 2D

X

Y

X'

Y'

P

P*

dX

dY


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Fig. 3.20 Scaling of geometry in 2D

X

Y

P

P*

X

sX

Y

sY


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Fig. 3.21 Reflection of geometry in 2DY

X25

25

Y

X

Y

X

28 28

-X

-Y

-X


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Fig. 3.22 Example for reflection transformation

X

Y

P

P*

y-y


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Fig. 3-23 Example for rotation transformation

X

Y

P

P*

PPr

O

α

x*x

yy*

θ


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Fig. 3.24 Example

10

8.6613

.66

5

3.66

8.66

30°

5

X

Y


37

3.4.5 Concatenation of transformations

• [P*] = [Tn] [Tn-1] [Tn-2] .. [T3] [T2] [T1]


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3.4.6 Homogeneous Representation

[ ]

=

=

11001001

1**

* yx

dYdX

yx

P


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Rotation about an arbitrary point

• Translate the point P to O, the origin of the axes system.

• Rotate the object by the given angle• Translate the point back to its original

position.


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Fig. 3.25 Rotation about an arbitrary point

X

Y

P

P*

PPr

O

A

r θ

dX

dY


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Reflection about an arbitrary line

• Translate the mirror line along the Y-axis such that the line passes through the origin, O

• Rotate the mirror line such that it coincides with the X-axis.

• Mirror the object through the X-axis.• Rotate the mirror line back to the original

angle with X-axis.• Translate the mirror line along the Y-axis

back to the original position.


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Fig. 3.26 Example for reflection transformation

about an arbitrary line

X

Y

P*

P

O

C

θ


43

3.5 3D Transformations

=

11000100010001

1***

zyx

dZdYdX

zyx


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3.6 Mathematics of Projection


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Fig. 3.27 The principle of projection

Object

Projecting plane

Projector

Drawing


46

Fig. 3.28 An object enclosed in a cube to

obtain various parallel projections

A

B

C

D

E

Z

X

Y


47

Fig. 3.29 Orthographic projection of an object

A

B

C

D

E

BOTTOM VIEW

FRONT VIEW RIGHT SIDE VIEWLEFT SIDE VIEW

TOP VIEW

REAR VIEW

A

B

CD E

Z

X

Y

X Y

Y

X

ZZ


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3.7 Clipping


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Fig. 3.30 Clipping of geometry for display

T

RL

B


50

Fig. 3.31 The 4-digit coding of the line end

points for clipping

1001 1000 1010

0001 0000 0010

0101 0100 0110


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Fig. 3.32 Identical line clipping of two different

geometries


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Fig. 3.33 Clipping produced for different

geometries by polygon clipping


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Fig. 3.34 Back-face removal using the face

normal and projecting ray

X

Y

Z

PN


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Summary• In order to display the graphical information, which is vectorial in

nature, it is necessary to convert it into raster format.• For converting lines into raster format, DDA algorithm is

simplest while Bresenham’s algorithm reduces the computations into integer format thereby making it a faster alternative.

• It is necessary to modify the pixel information for display to get more realistic visual experience.

• Depending upon the type of graphic display used, it is necessary to be familiar with a number of different coordinate systems to facilitate the graphic construction as well as display.

• In addition to the actual graphic information, a large amount ofadditional data such as organizational and technological data isstored with the product data.


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Summary• Geometric transformations can be handled conveniently using

matrix algebra. For this purpose it is necessary to use homogenous representation of vertex data.

• Various transformations that are quite useful are translation, rotation, scale and reflection. The 2D transformation methods can be easily extended into 3D.

• The 3D geometry data needs to be converted into 2D by adopting a suitable projection system such as orthographic, isometric or perspective projection.

• Since only part of the geometric model will be displayed most ofthe time, it is necessary to clip the information outside the display window.

• Also it is necessary some times to remove the hidden lines to make the display easier to understand. For this purpose back face removal and depth buffer (Z) are used.

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