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…