1. Computer Graphics Lecture 12 2D Transformations II Taqdees A
Computer Graphics Lecture 12 2D Transformations II Taqdees A. Siddiqi

2. Homogeneous Coordinates
Translation:
P′ = P + T
Rotation:
P′ = R . P
Scaling:
P′ = S . P

3. Homogeneous Coordinates
From our last discussion we know:
P’ = M1 . P + M2
With coordinate positions P and P’ represented as column vectors
Matrix M1 is a 2x2 array containing multiplicative factors
M2 is a 2 element column matrix containing translation factors

4. Can we find a way to eliminate the matrix addition associated with translation?
Yes, we can. But for that M1 will have to be rewritten as a 3x3 matrix and also the coordinate positions will have to be expressed as a homogeneous coordinate triple:
(x, y) as (xh, yh, h) where
xh yh
x = –––– , y = ––––
h h

5. For simplicity, 1 is used as value for h in affine transformations, thus
(x, y) has homogeneous coordinates as (x, y, 1)

6. Translation with Homogeneous Coordinates
The translation can now be expressed using homogeneous coordinates as:
Abbreviated as:
P’ = T (tx, ty) . P

7. Rotation with Homogeneous Coordinates
The rotation can now be expressed using homogeneous coordinates as:
Abbreviated as:
P’ = R (θ) . P

8. Scaling with Homogeneous Coordinates
The scaling can now be expressed using homogeneous coordinates as:
Abbreviated as:
P’ = S (Sx, Sy) . P

9. Composite Transformations
Two successive translations
Translation vectors (tx1, ty1) and (tx2, ty2)
P’ = T(tx2,ty2) . {T(tx1,ty1) . P}
= {T(tx2,ty2) . T(tx1,ty1)} . P
Where P and P’ are homogeneous column vectors.

10. Composite Translations
Which means that
T(tx2,ty2) . T(tx1,ty1) = T(tx1 + tx2 , ty1 + ty2)
hence

11. Composite Rotations
Two successive Rotations applied to a point P produce:
P’ = R(θ2) . {R(θ1) . P}
= {R(θ2) . R(θ1)} . P
But since,
R(θ2) . R(θ1) = R(θ1 + θ2)
Therefore,
P’ = R(θ1 + θ2) . P

12. Composite Scalings Two successive scaling operations would produce: or
S(sx2,sy2).S(sx1,sy1) = S(sx1.sx2, sy1.sy2)

13. General Pivot Point Rotation
General pivot point (xr,yr)
Translate-Rotate-Translate
Translate the object to coincide the pivot with the origin
Rotate about the origin
Translate back to the original pivot position

15. T(xr , yr) . R(θ) . T(-xr ,-yr) = R(xr, yr , θ)

16. General Fixed Point Scaling
General fixed point (xf,yf)
Translate-Scale-Translate
Translate the object to coincide the fixed point with the origin
Scale w.r.t. origin
Translate the object back to the original position

18. T(xf,yf).S(sx,sy).T(-xf,-yf) = S(xf,yf , sx,sy)

19. General Composite Transformations and Computational Efficiency
Translations-Rotations-Scalings

20. rsij are Rotation-Scaling terms
trsi are the translational terms
It takes 9 multiplications and 6 additions to evaluate but explicit calculations for transformed-coords:
x’ = x.rsxx + y.rsxy + trsx
y’ = x.rsyx + y.rsyy + trsy
i.e. 4 multiplications and 4 additions

21. If an object is to be scaled and rotated about its centroid coordinates (xc,yc) and then translated, the composite transformation matrix will be:
T(tx,ty) . R(xc,yc) . S(xc,yc,sx,sy)

22. T(tx,ty) . R(xc,yc) . S(xc,yc,sx,sy)

23. Order is Important

24. Reverse Order – Changed Position

25. Reflection Mirror image of object relative to axis of reflection
By rotating the object about axis of reflection by 180o

26. Reflection about x-axis

27. Reflection about y-axis

28. Shear Distorts the shape Effect is, of layers sliding off each other
x direction shear
y direction shear

29. Shear Y X 1 2 3 4 5 6 7 8 9 10

30. x direction Shear
x’ = x + shx . y
y’ = y

31. y direction Shear
x’ = x
y’ = shy . x + y

32. x and y direction Shear
x’ = x + shx . y
y’ = shy . x + y

33. Computer Graphics Lecture 12 2D Transformations II Taqdees A
Computer Graphics Lecture 12 2D Transformations II Taqdees A. Siddiqi