1. Computer Graphics 3: 2D Transformations

2. Contents In today’s lecture we’ll cover the following:
Why transformations
Transformations
Translation
Scaling
Rotation
Homogeneous coordinates
Matrix multiplications
Combining transformations

3. Why Transformations?
In graphics, once we have an object described, transformations are used to move that object, scale it and rotate it
Images taken from Hearn & Baker, “Computer Graphics with OpenGL” (2004)

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

5. Translation Example y x (2, 3) (1, 1) (3, 1) 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
(2, 3)
Translate by (5, 3)
x’ = x + dx
y’ = y + dy
(1, 1)
(3, 1)

6. Scaling 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. Scaling Example y (2, 3) (1, 1) (3, 1) 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
(2, 3)
Scale by (2, 2)
x’ = sx*x
y’ = sy*y
(1, 1)
(3, 1)

8. Rotation 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

9. Rotation Example y (4, 3) (3, 1) (5, 1) 1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
(4, 3)
xnew = xold × cosθ – yold × sinθ
ynew = xold × sinθ + yold × cosθ
1 radian = 180/π degrees
cos(π/6)=0.866
sin(π/6)=0.5
x’(4, 3) = 4*0.866 – 3*0.5 = 3.46 – 1.5 = 1.96
y’(4, 3) = 4* *0.866 = = 4.598
x’(3, 1) = 3*0.866 – 1*0.5 = – 0.5 = 2.098
y’(3, 1) = 3* *0.866 = = 2.366
x’(5, 1) = 5*0.866 – 1*0.5 = 4.33 – 0.5 = 3.83
y’(5, 1) = 5* *0.866 = = 3.366
(3, 1)
(5, 1)

10. 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)

11. 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!

12. 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:

13. Remember Matrix Multiplication
Recall how matrix multiplication takes place:

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

15. Homogenous Coordinates (cont…)
Rotation:

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

17. 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

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

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

20. Summary In this lecture we have taken a look at:
2D Transformations
Translation
Scaling
Rotation
Homogeneous coordinates
Matrix multiplications
Combining transformations
Next time we’ll start to look at how we take these abstract shapes etc and get them on-screen

21. Exercises 1 Translate the shape below by (7, 2) y (2, 3) (1, 2) (3, 2)1 2 3 4 5 6 7 8 9 10
(2, 3)
(1, 2)
(3, 2)
(2, 1) x

22. Exercises 2 Scale the shape below by 3 in x and 2 in y y (2, 3) (1, 2)1 2 3 4 5 6 7 8 9 10
(2, 3)
(1, 2)
(3, 2)
(2, 1) x

23. Exercises 3 Rotate the shape below by 30° about the origin y (7, 3)1 2 3 4 5 6 7 8 9 10
(7, 3)
(6, 2)
(8, 2)
(7, 1) x

24. Exercise 4
Write out the homogeneous matrices for the previous three transformations
Translation
Scaling
Rotation

25. Exercises 5
Using matrix multiplication calculate the rotation of the shape below by 45° about its centre (5, 3) x y 1 2 3 4 5 6 7 8 9 10
(5, 4)
(6, 3)
(4, 3)
(5, 2)

26. Scratch y 1 2 3 4 5 6 7 8 9 10 x

27. Equations xnew = xold + dx ynew = yold + dy Translation: Scaling:
xnew = Sx × xold ynew = Sy × yold
Rotation
xnew = xold × cosθ – yold × sinθ
ynew = xold × sinθ + yold × cosθ