By: Chandresh Suthar

1. GANDHINAGAR INSTITUTE OF TECHNOLOGY
Computer Aided Design (2161903)
Active Learning Assignment
On
Homogeneous representation in geometric transformation
Branch : Mechanical Engineering
Batch : 6 C-3
Prepared by: Guided By:
Suthar Chandresh (140120119229) Prof. Jatin Patel

2. Contents
• Definition & Motivation
• Geometric Transformation
:- Translation
:- Scaling
:- Shearing
• Matrix Representation
• Homogeneous Co-ordinates

3. Geometric transformation
• Definition
:- Translation , Scaling , Shearing
• Motivation – Why do we need geometric
transformations in CG?
:- As a viewing aid
:- As a modeling tool
:- As an image manipulation tool

4. Example: 2D Translation
Modeling
Coordinates
Translate(5, 3)
World Coordinates

5. EXAMPLE: 2D SCALING
Modeling
Coordinates
World Coordinates
Scale(0.3, 0.3)

6. Basic 2D Transformations
• Translation
–
–
• Scale
–
–
• Shear
–
–
txxx 
tyyy 
sxxx 
syyy 
yhxxx 
xhyyy 

7. Matrix Representation
• Represent a 2D Transformation by a Matrix
• Apply the Transformation to a Point




















y
x
dc
ba
y
x
dycxy
byaxx








dc
ba
Transformation
Matrix
Point

8. Matrix Representation
• Transformations can be combined by matrix
multiplication
































y
x
lk
ji
hg
fe
dc
ba
y
x
Matrices are a convenient and efficient way
to represent a sequence of transformations
Transformation
Matrix

9. 2×2 Matrices
• What types of transformations can be
represented with a 2×2 matrix?
2D Translation
txxx 
tyyy  



















y
x
ty
tx
y
x
0
0

10. 2×2 Matrices
• What types of transformations can be
represented with a 2×2 matrix?2D
Scaling
ysyy
xsxx






















y
x
sy
sx
y
x
0
0

11. 2×2 Matrices
• What types of transformations can be
represented with a 2×2 matrix?
2D Shearing




















y
x
shy
shx
y
x
1
1
yxshyy
yshxxx



12. Basic 2D Transformations
• Basic 2D transformations as 3x3 Matrices

































1100
10
01
1
y
x
ty
tx
y
x

































1100
00
00
1
y
x
sy
sx
y
x

































1100
01
01
1
y
x
shy
shx
y
x
Translate
Shear
Scale