1. Computer Graphic
2 D Transformation

2. Outline 1. Why transformations ? 2. Basic 2D Transformations
Translation
Scaling
Rotation
3. Homogeneous Coordinate system
4. Composite Transformations
5. Reflection and Shearing

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

4. Basic 2D Transformations
Translation
Simply moves an object from one position to another
Note: House shifts position relative to origin y x 1 2 3 4 5 6 7 8 9 10

5. 2D Translation
Moving a polygon from position (a) to position (b) with the translation vector (-5, 10), i.e. x y 5 10 15 20 x y 5 10 15 20
p3(15,10)
P1(10,5)
P2(20,5)
(a)
(b)

6. Basic 2D Transformations
Scaling
Scalar multiplies all coordinates
WATCH OUT: Objects grow and move!
Note: House shifts position relative to origin y x 1 2 3 4 5 6 7 8 9 10

7. Example: Non- Uniform Scaling
(b)
Non-uniform scaling

8. Basic 2D Transformations
Rotation
sin(a+b) = sin(a)cos(b) + sin(b)cos(a) 
cos(a+b) = cos(a)cos(b) - sin(a)sin(b) 
sin(a+b) = sin(a)cos(b) + sin(b)cos(a) 
cos(a+b) = cos(a)cos(b) - sin(a)sin(b) 
Similarly,

9. Rotation

10. Homogenous 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 Homogenous 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. Example: Rotate a polygonal object defined by vertices A(0,0),B(1,0) ,C(1,1) and D(0,1) by 45 about the origin

14. Points are always rotated about the origin
Rotation
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

15. 2D Transformations in Homogenous Notation
Rotation
Scaling
Translation

16. Inverse Transformations
Inverse of Rotation
Inverse of Scaling
Inverse of Translation

17. Example
Consider the line with endpoints (10, 10) and (30, 25). Translate it by tx = -20, ty = -10 and then rotate it by θ = 90º. x y
(10, 10)
(30, 25)
Right-to-left

18. Solution x y
(10, 10)
(30, 25)

19. Solution (continue) Point (10, 10) Point (30, 25) y x (30, 25)
(0, -10)
(-15, 10)
Point (10, 10)
Point (30, 25)

20. Exercises Consider the following object:
Apply a rotation by 145º then scale it by Sx=2 and Sy=1.5 and then translate it by tx=20 and ty=-30.
Scale it by Sx=½ and Sy=2 and then rotate it by 30º.
Apply a rotation by 90º and then another rotation by 45º.
Apply a rotation by 135º. x y 10 25 45

21. Scaling about a Reference Point
If the center of the object is not at the origin, in this case we should consider a scaling as a composite transformation which involves the following steps.
Translation of the reference (xc,yc) to the origin
Scale the object by scale factor by (Sx,Sy)
Translation of the reference point back to its original position

22. (a) Original position of object and fixed point
(xc, yc)T
(a) Original position of
object and fixed point
(b) Translate object so that
fixed point (xr, yr) is at origin
(xc, yc)T
(c) Scale object with
respect to origin
(d) Translate object so that
fixed point is returned to position (xr, yr)T 22

23. Combining the Transformation
The three transformation matrices are combined as follows

24. Example1:
Consider a triangle having a vertices at A(0,0), B(5,1), and C(3,4) scale this triangle using scaling factors Sx=2 and Sy=1 about a) the origin b) about the reference point (2,3)
a) At origin ( xr=0, yr=0)

25. b) Scaling about the reference point (2,3) : Here, the transformation matrix

26. Rotation about an Arbitrary Point
Rotation about an arbitrary point is composite transformation . This operation involve s three basic transformations:
Translation of the object center (xc,yc) to the origin
Rotation of the object about the origin by angle θ in.say, the anticlockwise direction
Translation of the rotated object back to its original position

27. 1 2 3 4

28. Example: Consider a triangle having a vertices at A(0,0), B(5,1), and C(3,4) scale this triangle. Rotate the triangle by
90 about a) the origin b) about the reference point (2,3)
Solution: a) about the origin
The transformation matrix for rotation about the origin :

29. b) Rotation about point (2,3 )

31. Exercises Consider the following object:
Apply a rotation by 60° on the Pivot Point (-10, 10) and display it.
Apply a rotation by 30° on the Pivot Point (45, 10) and display it.
Apply a rotation by 270° on the Pivot Point (10, 0) and then translate it by tx = -20 and ty = 5. Display the final result. x y 10 25 45

32. Reflection About the x axis About the y axis y x y x
Reflection of an object about the x axis x y
About the y axis x y
Reflection of an object about the y axis

33. Same as a rotation with 180º
Reflection
Relative to the coordinate origin
Same as a rotation with 180º x y
With respect to the line y = x x y
y = x

34. Reflection about x=-y line : the x and y coordinates are interchanged and their values are negated after reflection
Original
position y
y = -x 3 1 2
Reflected
position 1’ 3’ 2’ x

35. Reflection about any arbitrary line: y=mx+b
This kind of reflection can be implemented by a sequence of the following steps:
Translate the line, along the y-axis so that it pass the origin
Rotate through an angle θ =tan-1(m) in clockwise so that the line is aligned with x-axis
Reflect the object about x-axis
Perform inverse Rotation and Translation

36. Example: Consider a triangle whose vertices are (2 2), (4 2) and (4 4)
Example: Consider a triangle whose vertices are (2 2), (4 2) and (4 4). Find the concatenated transformation matrix and the transformed vertices for rotation of 90 about the origin followed by reflection through the line y = -x. Comment on the sequence of transformations.

37. Ex: Reflect triangle (2,4),(4,6),(2,6) about line y=0.5(x+4)
Translate (0, -2) so that the line passes through the origin b ϴ
Rotate the line about the x axis by -θo

38. Reflect object about the x axis
Rotate back the line by θo

39. 5. Translate back (0,2)

40. The final position of line after reflection about the line is

41. Shearing about the originy y y
(1, 3)
(1, 2)
(1, 1)
(2, 1)
(3, 1)
(0, 1)
(0, 1) x x x
(0, 0)
(1, 0)
(0, 0)
(1, 0)
(0, 0)
(a)
(b)
(c)
SHy=2
SHx=2 41

42. Shearing about the a reference point ( xr,yr)

43. Shear Example