homogeneous representation in geometric transformation
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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
Contents
• Definition & Motivation
• Geometric Transformation
:- Translation
:- Scaling
:- Shearing
• Matrix Representation
• Homogeneous Co-ordinates
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
Example: 2D Translation
Modeling
Coordinates
Translate(5, 3)
World Coordinates
EXAMPLE: 2D SCALING
Modeling
Coordinates
World CoordinatesScale(0.3, 0.3)
Basic 2D Transformations
• Translation–
–
• Scale–
–
• Shear–
–
txxx
tyyy
sxxx
syyy
yhxxx
xhyyy
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
MatrixPoint
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
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
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
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
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
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