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