1. Lecture Notes:
Computer Graphics

2. Why Transformations?
In graphics, once we have an object described, transformations are used to move that object, scale it and rotate it

3. Transformation Two complementary views
Geometric trans: Object is transformed relative to a stationary coordinate system
Coordinate trans: Object is held stationary while the coordinate system is transformed
Ex: moving an automobile against a scenic background
Move the auto and keep the background fixed (GT)
Keep the car fixed and move the background (CT)

4. Basic 2D Geometric Transformations
Four basic types
Translation
Scaling
Rotation
Shear (possible using a combination of rotation and scaling)

5. Translation xnew = xold + tx ynew = yold + ty
Simply moves an object from one position to another
xnew = xold + tx ynew = yold + ty
Note: House shifts position relative to origin y x 1 2 3 4 5 6 7 8 9 10

6. Scaling xnew = Sx × xold ynew = Sy × yold
Scalar multiplies all coordinates
WATCH OUT: Objects grow and move!
xnew = Sx × xold ynew = Sy × yold
Note: House shifts position relative to origin y x 1 2 3 4 5 6 7 8 9 10

7. Rotation xnew = xold × cosθ – yold × sinθ
Rotates all coordinates by a specified angle
xnew = xold × cosθ – yold × sinθ
ynew = xold × sinθ + yold × cosθ
Points are always rotated about the origin y 6 5 4 3 2 1 x 1 2 3 4 5 6 7 8 9 10

8. Shear
Along X-axis
Along Y-axis

9. Basic 2D Geometric Transformations
Translation
Scale
Rotation
Shear

10. Transformation and Matrix Multiplication
Recall how matrix multiplication takes place:

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

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

13. 2×2 Matrices
What types of transformations can be represented with a 2×2 matrix?
2D Identity
2D Scaling

14. 2×2 Matrices
What types of transformations can be represented with a 2×2 matrix?
2D Rotation
2D Shearing

15. 2×2 Matrices 2D Mirror over Y axis
What types of transformations can be represented with a 2×2 matrix?
2D Mirror over Y axis
2D Mirror over (0,0)

16. 2×2 Matrices 2D Translation
What types of transformations can be represented with a 2×2 matrix?
2D Translation
NO!!
Only linear 2D transformations
can be Represented with 2x2 matrix

17. Homogeneous Coordinates
A point (x, y) can be re-written in homogeneous coordinates as (xh, yh, h)
The homogeneous parameter h is a non-zero value such that:
We can then write any point (x, y) as (hx, hy, h)
We can conveniently choose h = 1 so that (x, y) becomes (x, y, 1)

18. Homogeneous Coordinates
Add a 3rd coordinate to every 2D point
(x, y, w) represents a point at location (x/w, y/w)
(x, y, 0) represents a point at infinity
(0, 0, 0) is not allowed y 2
(2, 1, 1) or (4, 2, 2) or (6, 3, 3) 1 x 1 2
Convenient Coordinate System to
Represent Many Useful Transformations

19. Why Homogeneous Coordinates?
Mathematicians commonly use homogeneous coordinates as they allow scaling factors to be removed from equations
We will see in a moment that all of the transformations we discussed previously can be represented as 3*3 matrices
Using homogeneous coordinates allows us use matrix multiplication to calculate transformations – extremely efficient!

20. Homogeneous Translation
The translation of a point by (dx, dy) can be written in matrix form as:
Representing the point as a homogeneous column vector we perform the calculation as:

21. Homogenous Coordinates
To make operations easier, 2-D points are written as homogenous coordinate column vectors
Translation:
Scaling:

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

23. Combining Transformations
A number of transformations can be combined into one matrix to make things easy
Allowed by the fact that we use homogenous coordinates
Imagine rotating a polygon around a point other than the origin
Transform to centre point to origin
Rotate around origin
Transform back to centre point

24. Matrix Composition
Transformations can be combined by matrix multiplication
Efficiency with premultiplication
Matrix multiplication is associative

25. Combining Transformations (cont…)1 2 3 4

26. Combining Transformations (cont…)
The three transformation matrices are combined as follows
REMEMBER: Matrix multiplication is not commutative so order matters

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

28. Inverse Transformations
Transformations can easily be reversed using inverse transformations

29. Basic 2D Coordinate Transformations
Translation
Scale
Rotation
Shear ??