Lecture Computer graphics: Lecture 15 - Fasih ur Rehman

This chapter discuss the purpose of the components required for successful communications; describe these uses of computer communications: wireless messaging services, wireless Internet access points, cybercafés, global positioning systems, collaboration, groupware, voice mail, and Web services;

Computer Graphics

Lecture 15

Fasih ur Rehman

Last Class

* Combining Transformations

- Affine versus Rigid body Transformations

* Homogenous Transformations

Today’s Agenda

Homogeneous transformations

- Types of Transformations

— Linear Transformations

— Affine Transformations

— Projective Transformations

Homogenous Coordinate

- Basic 2D transf@/Sl©fBin 3D are

— Translation

x 1 O a

y 0 1 b

1 0 0 1 —

scaling as 3x3

a OO x

O b OO y

QO O 1

x'

Rotation as 3x3

x COs sin QO x

y sin COS Q y

l 0 0 1 |

Shear as 3x3

| h O x

|

O O |

|

Linear Transformations

* Combination of Scaling, rotation and shear

are linear transformations

!

!

y cd y

- Linear Transformations satisfy following

—- Origin maps origin

— Lines maps lines

Affine Transformations

- Affine transformations are combinations of

linear transformations and Translation

y deféey

- Affine transformations obey the following

—- Origin does not necessarily map the origin

— Lines maps lines

Projective Transformations

- Affine Transformations and projective warps form projective transformation

y defy

W g h i w

- Projective Transformations obey the

following

— Origin does not necessarily map the origin

Matrix Composition

‘ Homogenous transformations can also be combined by Matrix multiplication

Example

Multiplication Order

* Scaling, Rotation and then translation

Summary

Homogeneous transformations

- Types of Transformations

— Linear Transformations

— Affine Transformations

— Projective Transformations

References

‘ Fundamentals of Computer Graphics Third Edition by Peter Shirley and Steve

Marschner

* Interactive Computer Graphics, A Top-

down Approach with OpenGL (Sixth

Edition) by Edward Angel