CAD/CAM, Unit-I, SPPU, BE Mechanical, Geometric Transformations in CAD, scaling, rotation, move, formulation, translation, reflection, concatenated transformation, homogeneous transformation, mirror command in CAD, PROJECTIONS OF GEOMETRIC MODELS,

1. Chapter 1
computer graphics
S.G.R. Education Foundation
G H RAISONI COLLEGE OF ENGINEERING AND MANAGEMENT
AHMEDNAGAR
NAAC ACCREDITED
(APPROVED BY AICTE, NEW DELHI, RECOGNIZED BY GOVT. OF MAHARASHTRA & AFFILIATED TO SAVITRIBAI PHULE PUNE UNIVERSITY)
- Prof. Aniket V. Joshi
Asst. Professor
Mechanical Engineering Department
GHRCEM, A’Nagar.

2. Introduction
 CAD: use of computer to assist the user in design of system.
Stages in CAD:
Create
Modify
Analyze
Optimize
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3. Introduction
Interactive Computer
Graphics:
o Displays data and information in
the form of graphics
o Data entered is get converted into
graphical form by using software
and hardware
o User can create, modify and
explore further possibilities and
options
o Major advantage is change in the
image
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4. GEOMETRIC TRANSFORMATION
TRANSLATION
ROTATION
SCALING
REFLECTION
SHEAR
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5. GEOMETRIC TRANSFORMATION
 Transformations: changes performed on the original
graphic image by changing database.
 Used to alter the orientation, scale, position of the drawing
 Applications of Transformations:
a. Must be creation of model
b. To express location of objects relative to others
c. To view an object from different positions and directions
d. To perform transformations like translate/move, rotate,
scale, mirror, etc..
e. To obtain orthographic and perspective views of model
f. To create animation
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6. Formulation
• Point can be represented as P= [x,y]
• Line can be represented as L=
• L’=L [TM] where [TM]= Transformation Matrix
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7. translation
o Entity of a geometric model remains parallel to its initial
position
o Every point on geometric model moves by equal distance
T = [tx ty]
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A’B’C’D’ = ABCD + T

8. translation
 Translation can be applied to curves such as circles,
parabolas, surfaces and solids. These shapes are treated as
a geometric entity.
 Limitations for translation of circle:
a) Circle would be broken down into finite points and
transformations for each of these points would then have
to be done
b) Resulting entity would lose its identity of being a circle.
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9. rotation
 Turning the object through angle ‘θ’ about the origin (@ z
axis).
 Used to view object from different angles.
 Allows the user to create an array of objects.
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A’B’ = AB * R

10. scaling
 Alters the size of an object.
 It can be Uniform (equal in both X and Y directions) or non-
uniform (different in X and Y directions)
P’ = P * S
[x’ y’] = [x y]
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11. Reflection (mirror)
 Process of obtaining a mirror of the original shape
 Used in symmetrical objects
P’= P*M
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12. shear
• Causes the image to slant.
P’ = P*SH
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13. Homogeneous co-ordinates
 Need:
o For quick and efficient calculations, it is desirable to express
all transformations in the form of multiplications.
o To represent points at infinity and non intersection of
parallel lines.
o To represent multiple operations which include translation.
o To draw perspective views of geometrical models
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14. Homogeneous co-ordinates
• In homogeneous coordinate system, point P(x, y) can be
expressed as P(x’ y’ h)
Where
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 h is non zero number, convenient value is 1
General 3x3 matrix for
homogeneous
transformation.
a d 0
b e 0
c f 1

15. Translation
matrix in
homogeneous
form
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16. Inverse transformations
• Inverse of matrix is another matrix such that when two are
multiplied, an identity matrix results. i.e. T*T-1 = I
• For point p, p’ is transformed point
Then P’= P*T
P’*T-1 = P*T* T-1
P’*T-1 = P*I
P’*T-1 = P
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17. Concatenated or composite
transformation
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18. MAPPING OF GEOMETRIC
MODELS
 Types of coordinate system:
1) Model Coordinate System/ World Coordinate System:
 Reference space of the model is stored with respect to all
geometrical data
 Cartesian coordinate system
 Coordinates also referred
as global coordinates
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19. MAPPING OF GEOMETRIC MODELS
2) Local Coordinate System/ User Coordinate System/ Working
Coordinate System:
 Convenient to use
 All the coordinates depend upon the origin of the model
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20. Three dimensional
transformations
 The transformation takes place on a 3D plane
 Generalize from 2D by including z coordinate
 Straight forward for translation and scale, rotation
 more difficult
 Homogeneous coordinates: 4 components
 Transformation matrices: 4×4 elements
a b c tx
d e f ty
g h I tz
0 0 0 1
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21. Projections of geometric
models
 3D objects need to be displayed in 2D form
 Types of projections are:
1) Parallel projection
2) Perspective projection
1) Parallel Projection
-Centre of projection is
taken at infinity.
-Parallel projections preserve
The parallelism
-Method is used to generate orthographic views
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Parallel Projection

22. Projections of geometric
models
2) Perspective Projection
- Centre of projection is a point at finite distance from the
object.
- Create artistic views and used by architects
- Actual dimension and angles cannot preserve on the
drawing
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