1. 2D Geometric Transformations
Chapter 4
2D Geometric Transformations

2. Agenda Definition & Motivation 2D Geometric Transformation
Translation
Rotation
Scaling
Matrix Representation
Homogeneous Coordinates
Matrix Composition
Composite Transformations
Pivot-Point Rotation
General Fixed-Point Scaling
Reflection and Shearing
Transformations Between Coordinate Systems

3. Geometric Transformation
Definition
Translation, Rotation, Scaling
Motivation – Why do we need geometric transformations in CG?
As a viewing aid
As a modeling tool
As an image manipulation tool

4. Basic 2D Transformations
Translation
Scale
Rotation
Shear

5. Transformation Matrix Notation Inverse Transformation Matrix
Matrix Notations
Transformation Name
Transformation Matrix Notation
Inverse Transformation Matrix
Translation
T(a, b)
T-1(a,b) = T(-a, -b)
Rotation
R(θ)
R-1(θ) = R(-θ)
Scaling
S(sx, sy)
S-1 (sx, sy) =( 1 𝑆𝑥 , 1 𝑆𝑦 )
Reflection About x-axis
Mx(a, b)
Mx-1 (a, b) = Mx(a, -b)
Reflection About y-axis
My(a, b)
My-1 (a, b) = My(-a, b)
Reflection About the origin
Mo(a, b)
Mo-1 (a, b) = Mo(-a, -b)

6. Matrix Representation
Represent a 2D Transformation by a Matrix
Apply the Transformation to a Point
Transformation
Matrix
Point

7. Matrix Representation
Transformations can be combined by matrix multiplication
Transformation
Matrix
Matrices are a convenient and efficient way
to represent a sequence of transformations

8. 2×2 Matrix (2D Scaling)
2D Identity
2D Scaling

9. 2×2 Matrix (2D Rotation & Shearing)
2D Shearing

10. 2×2 Matrix (2D Reflection)
2D Mirror over Y axis
2D Mirror over (0,0)

11. 2×2 Matrix (2D Translation)
This transformation cannot be represented with a 2×2 matrix?
2D Translation
NO!!
Only linear 2D transformations
can be Represented with 2x2 matrix

12. Homogeneous coordinates
Reason?
Unification and extension of concepts
Best way to represent a sequence of transformations
2D translation can be represented by a 3×3 matrix
Point represented with homogeneous coordinates

13. Basic 2D Translation
A translation moves all points of an object a fixed distance in a specified direction.

14. Basic 2D Rotation
It rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system.

15. Basic 2D Scaling
It is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

16. Basic 2D Shearing
A transformation that distorts the shape of an object.
Two types of shearing transformations are there about X values and about Y values

17. Basic 2D Transformations
Basic 2D transformations as 3x3 Matrices
Translate
Scale
Rotate
Shear

18. Matrix Composition
Transformations can be combined by matrix multiplication
Matrix multiplication is associative

19. Matrix Composition Rotate by  around arbitrary point (a,b)
Scale by sx, sy around arbitrary point (a,b)
(a,b)
(a,b)

20. Pivot-Point Rotation
(xr,yr)
Translate
Rotate
Translate

21. General Fixed-Point Scaling
(xf,yf)
Translate
Scale
Translate

22. Translate-Rotate-Translate
Final position
Original position

23. Reflection Reflection about x-axis (y=0) Reflection about y-axis (x=0)
Refection about the origin (0, 0)
Reflection about the line y = x
Reflection about the line y = ax + b

24. Reflection about x-axis (y=0)1 x 3 2 1’ 3’ 2’

25. Reflection about y-axis (x=0)1 3 2 1’ 3’

26. Refection about the origin (0, 0)x y 3 1’ 3’ 2 1

27. Reflection about the line y = x
Reflection with respect to a Line
Clockwise rotation of 45  Reflection about the x axis  Counterclockwise rotation of 45 x y x y x y 1 3 2 1’ 3’ 2’ x y

28. Reflection about the line y = -x

29. Reflection about the line y = ax+b
Translation
Rotation
Reflection
Reverse Rotation
Reverse Translation x y