1. 2-D Geometry

2. The Image Formation Pipeline

3. Outline Vector, matrix basics 2-D point transformations
Translation, scaling, rotation, shear
Homogeneous coordinates and transformations
Homography, affine transformation

4. Notes on Notation (a little different from book)
Vectors, points: x, v (assume column vectors)
Matrices: R, T
Scalars: x, a
Axes, objects: X, Y, O
Coordinate systems: W, C
Number systems: R, Z
Specials
Transpose operator: xT (as opposed to x0)
Identity matrix: Id
Matrices/vectors of zeroes, ones: 0, 1

5. Block Notation for Matrices
Often convenient to write matrices in terms of parts
Smaller matrices for blocks
Row, column vectors for ranges of entries on rows, columns, respectively
E.g.: If A is 3 x 3 and :

6. 2-D Transformations Types Mathematical representation Scaling Rotation
Shear
Translation
Mathematical representation

7. 2-D Scaling

8. 2-D Scaling

9. 2-D Scaling sx 1
Horizontal shift proportional to horizontal position

10. 2-D Scaling sy 1
Vertical shift proportional to vertical position

11. 2-D Scaling

12. Matrix form of 2-D Scaling

13. 2-D Scaling

14. 2-D Rotation

15. 2-D Rotation

16. 2-D Rotation

17. Matrix form of 2-D Rotation
(this is a counterclockwise rotation; reverse signs of sines to get a clockwise one)

18. Matrix form of 2-D Rotation

19. 2-D Shear (Horizontal)

20. 2-D Shear (Horizontal)
Horizontal displacement proportional to vertical position

21. (Shear factor h is positive for the figure above)
2-D Shear (Horizontal)
(Shear factor h is positive for the figure above)

22. 2-D Shear (Horizontal)

23. 2-D Shear (Vertical)
(v is negative for the figure above)

24. 2-D Translation

25. 2-D Translation

26. 2-D Translation

27. 2-D Translation

28. 2-D Translation

29. 2-D Translation

30. Representing Transformations
Note that we’ve defined translation as a vector addition but rotation, scaling, etc. as matrix multiplications
It’s inconvenient to have two different operations (addition and multiplication) for different forms of transformation
It would be desirable for all transformations to be expressed in a common, linear form (since matrix multiplications are linear transformations)

31. Example: “Trick” of additional coordinate makes this possible
Old way:
New way:

32. Translation Matrix
We can write the formula using this “expanded” transformation matrix as:
From now on, assume points are in this “expanded” form unless otherwise noted:

33. Homogeneous Coordinates
“Expanded” form is called homogeneous coordinates or projective space
Technically, 2-D projective space P2 is defined as Euclidean space R3 ¡ (0, 0, 0)T
Change to projective space by adding a scale factor (usually but not always 1):

34. Homogeneous Coordinates: Projective Space
Equivalence is defined up to scale, (non-zero for finite points)
Think of projective points in P2 as rays in R3, where z coordinate is scale factor
All Euclidean points along ray are “same” in this sense

35. Leaving Projective Space
Can go back to non-homogeneous representation by dividing by scale factor and dropping extra coordinate:
This is the same as saying “Where does the ray intersect the plane defined by z = 1”?

36. Homogeneous Coordinates: Rotations, etc.
A 2-D rotation, scaling, shear or other transformation normally expressed by a 2 x 2 matrix R is written in homogeneous coordinates with the following 3 x 3 matrix:
The non-commutativity of matrix multiplication explains why different transformation orders give different results—i.e., RT  TR
In homogeneous
form
In homogeneous form

37. Example: Transformations Don’t Commute

38. Example: Transformations Don’t Commute

39. Example: Transformations Don’t Commute

40. Example: Transformations Don’t Commute

41. 2-D Transformations
Full-generality 3 x 3 homogeneous transformation is called a homography

42. Translation components
2-D Transformations
Full-generality 3 x 3 homogeneous transformation is called a homography
Translation components

43. Scale/rotation components
2-D Transformations
Full-generality 3 x 3 homogeneous
transformation is called a homography
Scale/rotation components

44. Shear/rotation components
2-D Transformations
Full-generality 3 x 3 homogeneous transformation is called a homography
Shear/rotation components
Shear/rotation off-diagonal (more about relationship between shearing and rotation at

45. Homogeneous scaling factor
2-D Transformations
Full-generality 3 x 3 homogeneous transformation is called a homography
Homogeneous scaling factor

46. 2-D Transformations
Full-generality 3 x 3 homogeneous transformation is called a homography
When these are zero (as they have been so far), H is an affine transformation