This document discusses homogeneous representations of geometric transformations using matrices. It defines translation, scaling, and shearing transformations and explains how they can be represented by 2x2 and 3x3 matrices. Basic 2D transformations like translation, scaling, and shearing can be modeled as linear transformations and combined using matrix multiplication, allowing geometric operations to be expressed through matrix algebra. Homogeneous coordinates allow geometric transformations to be uniformly represented by matrices.

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