1. Computer Graphics
Transformations

2. Types of Transformations
Geometric Transformations
Translation
Rotation
scaling
Linear (preserves parallel lines)‏
Non-uniform scales, shears or skews
Projection (preserves lines)‏
Perspective projection
Parallel projection
Non-linear (lines become curves)‏
Twists, bends, warps, morphs,

3. 2D Transforms What am I talking about when I say “transforms”? (x’,y’)
• Translation
• Scaling
• Rotation

4. 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)

5. 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

6. 2D Translations. P’ P

7. 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)

8. 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

9. Scaling
Scaling a coordinate means multiplying each of its components by a scalar
Uniform scaling means this scalar is the same for all components:
 2

10. Scaling Non-uniform scaling: different scalars per component:
How can we represent this in matrix form?
X  2, Y  0.5

11. 2D Scaling from the origin.P’ P

12. 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)

13. 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

14. 2D Rotation about the origin.
Rewriting in matrix form gives us :

15. 2D Rotation about the origin.y
P’(x’,y’)
P(x,y) r  y r  x x

16. 2D Rotation about the origin.y
P’(x’,y’)
P(x,y) r  y r  x x

17. 2D Rotation about the origin.
Substituting for r :
Gives us :

18. Rotation About a Point: Moving Object
rotate about
p by :
translate p
to origin
rotate about
origin
translate p
back FW

19. Rotation Example Two transforms: Scale x and y by a factor of 2
Translate points (+3, +2)
Rotation around (1, 2) by angle 45o y 1 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)

20. Homogeneous Coordinates
Homogeneous representation of points:
Add an additional component w=1 to all points
All multiples of this vector are considered to represent the same 3D point
why bother? need unified representation

21. Homogeneous Coordinates
represent coordinates in 2 dimensions with a 3-vector
Homogeneous coordinates seem unintuitive, but they make graphics operations much easier

22. Geometrically In 2D
Cartesian Coordinates: y x

23. Geometrically In 2D
Homogeneous Coordinates: w
w=1 y x

24. 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)

25. 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!

26. Homogeneous Coordinates
Q: How can we represent translation as a 3x3 matrix?
A: Using the rightmost column:

27. Basic Transformations: Scaling
 2
X  2, Y  0.5
Scaling about the origin

28. Basic Transformations: Rotation P Q
Qx = Px cos() - Py sin()
Qy = Px sin() + Py cos()
Rotation about the origin

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

30. 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

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

32. Composing of Affine Transformations
Example: Rotation around arbitrary center

33. Composing of Affine Transformations
Example: Rotation around arbitrary center
Step 1: translate coordinate system to rotation center

34. Composing of Affine Transformations
Example: Rotation around arbitrary center
Step 2: perform rotation

35. Composing of Affine Transformations
Example: Rotation around arbitrary center
Step 3: back to original coordinate system

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

37. Two Transform Paths
Scale then Translate

38. Concatenation of scales.

39. Non-commutative Composition
Rotate then Translate: p' = T ( R p ) = TR p 1 1 3 1 1 1 1 -1 -1 -1 1 1 1 -1 3 1
TR = =
Translate then Rotate: p' = R ( T p ) = RT p 1 1 1 -1 -1 -1 1 1 1 1 1 1 3 1 1 -1 -1 3 1
RT = =
02/10/09
Lecture 4

40. 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

41. 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

42. 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

43. 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)

44. This is convenient for character animation / robotics
In robotics / animation, we often want to know what is the current 3D location of the end effectors (like the hand)‏
Can concatenate matrices from the origin of the body towards the end effecter

45. 3D Transformations. Use homogeneous coordinates, just as in 2D case.
Transformations are now 4x4 matrices.
We will use a right-handed (world) coordinate system - ( z out of page ).
z (out of page)‏ y x

46. Translation in 3D.
Simple extension to the 3D case:

47. 3D Scaling
Original
scale Y axis
scale all axes

48. 3D Rotations
Rotation about z-axis (z-roll)

49. Rotating About the x-axis Rx()‏

50. Rotating About the y-axis Ry()‏

51. Basic 3D Transformations
Rotate around Z axis:
Rotate around Y axis:
Rotate around X axis:

52. Composed Transformations
Rotation about an axis y u Q β P Φ θ x z
Ru(β) = Ry(-θ)*Rz(Φ)*Rx(β)*Rz(-Φ)*Ry(θ)

53. Transform Left-Right, Right-Left
Transforms between world coordinates and viewing coordinates. That is: between a right-handed set and a left-handed set.

54. Shearing

55. Shearing Horizontal Shear y x s=0, No Shear s=1, 45 Degree Shear
Vertical Shear x y
Pause to calculate
s is -tan(θ), where θ is the desired shear angle

56. Mirror Reflection about an Axis
If either x or y axis is treated as a mirror, the object has a mirror image or reflection
Mirror reflection transformation Mx about the x axis is given by
P = Mx (P)
Where x’ = x and y’ = -y
Mirror reflection transformation My about the y axis is given by
P = My (P)
Where x’ = -x and y’ = y
P(-x,y)
P(x,y)
P(x,-y)

57. Calculating the world coordinates of all vertices
For each object, there is a local-to-global transformation matrix
So we apply the transformations to all the vertices of each object
We now know the world coordinates of all the points in the scene

58. OpenGL Transformations
glTranslate{fd}(TYPE x, TYPE y, TYPE z)
Multiply the current matrix by the translation matrix
glRotate{fd}(TYPE angle, TYPE x, TYPE y, TYPE z)
Multiply the current matrix by the rotation matrix that rotates an object about the axis from (0,0,0) to (x, y, z)
glScale{fd}(TYPE x, TYPE y, TYPE z)
Multiply the current matrix by the scale matrix
Graphics Hardware
Vector/Raster Graphics