Homogeneous representation
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Transcript of Homogeneous representation
HOMOGENEOUS REPRESENTATION ; TRANSLATION ; SCALING ; SHEARING
Mechanical 6th-ASub :- CAD
Gohel Utsav (ME1319) Gosaliya Dheirya (ME1320)
CONTENTS
• Defination & Motivation• Geometric Transformation
:- Translation:- Scaling:- Shearing
• Matrix Representaion• Homogeneous Co-ordinates
GEOMETRIC TRANSFORMATION
• Defination:- 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
ModelingCoordinates
Translate(5, 3)
World Coordinates
EXAMPLE: 2D SCALING
ModelingCoordinates
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
yx
dcba
yx
dycxybyaxx
dcba
TransformationMatrix Point
MATRIX REPRESENTATION
• Transformations can be combined by matrix multiplication
yx
lkji
hgfe
dcba
yx
Matrices are a convenient and efficient wayto represent a sequence of transformations
TransformationMatrix
2×2 MATRICES
• What types of transformations can be represented with a 2×2 matrix?
2D Translation
txxx tyyy
yx
tytx
yx
00
2×2 MATRICES
• What types of transformations can be represented with a 2×2 matrix?2D
Scaling
ysyyxsxx
yx
sysx
yx
00
2×2 MATRICES
• What types of transformations can be represented with a 2×2 matrix?
2D Shearing
yx
shyshx
yx
11
yxshyyyshxxx
BASIC 2D TRANSFORMATIONS
• Basic 2D transformations as 3x3 Matrices
11001001
1yx
tytx
yx
11000000
1yx
sysx
yx
11000101
1yx
shyshx
yx
Translate
Shear
Scale
Thank You…