1. Two-Dimensional Geometric Transformations
Chapter 5
Two-Dimensional Geometric Transformations

2. Two-Dimensional Geometric Transformations
Basic Transformations
Translation
Rotation
Scaling
Composite Transformations
Other transformations
Reflection
Shear

3. Translation Translation transformation Rigid-body transformation y y
Translation vector or shift vector T = (tx, ty)
Rigid-body transformation
Moves objects without deformation y y P’ T T p x x

4. Rotation Rotation transformation x’=rcos(+)= rcoscos-rsin sin y
y’=rsin(+)= rcossin+rsin cos y
x=rcos y=rsin 
P’ (x’,y’)
x’=x cos-ysin
y’=xsin+ycos
P(x,y)   r x
P’= R· P

5. Rotation Pivot point x’=xr+(x- xr)cos-(y- yr)sin y
y’=yr+(x- xr)sin+(y- yr)cos y
P’ (x’,y’)
P(x,y)   r
(xr,yr) x

6. Scaling Scaling transformation Scaling factors, sx and sy
Uniform scaling y y x x

7. Scaling
Fixed point

8. Matrix Representations and Homogeneous Coordinates
Translation

9. Matrix Representations
Scaling
Rotation

10. Composite Transformations
Translations

11. Composite Transformations
Scaling

12. Composite Transformations
Rotations

13. General Pivot-Point Rotation
Rotations about any selected pivot point (xr,yr)
Translate-rotate-translate

14. General Pivot-Point Rotation

15. General Fixed-Point Scaling
Scaling with respect to a selected fixed position (xf,yf)

16. General Fixed-Point Scaling
Translate-scale-translate

17. General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions.
We scale an object in other directions with scaling factors s1 and s2

18. General Scaling Directions

19. Concatenation Properties
Matrix multiplication is associative.
A·B ·C = (A·B )·C = A·(B ·C)
Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated.
Except for some special cases:
Two successive rotations
Two successive translations
Two successive scalings
rotation and uniform scaling

20. Concatenation Properties
Reversing the order
A sequence of transformations is performed may affect the transformed position of an object.

21. General Composite Transformations and Computer Efficiency
A general two-dimensional transformation
Rotation-scaling terms rsij
Translational terms trsx and trsy
Minimum number of computations
Four multiplications
Four additions

22. Rigid-Body Transformation
Rigid-body transformation matrix
The upper-left 2-by-2 submatrix is an orthogonal matrix
Two vectors (rxx, rxy) and (ryx, ryy) form an orthogonal set of unit vectors.
Multiplicative rotation terms rij
Translational terms trx and try

23. Rigid-Body Transformation
The orthogonal property of rotation matrices
We know the final orientation of an object
Construct the desired transformation by assigning the elements of u’ to the first row of the rotation matrix and the elements of v’ to the second row.

24. Computational Efficiency
Use approximations and iterative calculations to reduce computations
Approximate the trigonometric functions based on the first few terms of their power-series expansions.
For small enough angles (< 100), cos is approximately 1.sin is approximately 
Accumulated error control
Estimate the error in x’ and y’ at each step
Reset object positions when the error accumulation becomes too great

25. Reflection A transformation produces a mirror image of an object.
Axis of reflection
A line in the xy plane
A line perpendicular to the xy plane
The mirror image is obtained by rotating the object 1800 about the reflection axis.
Rotation path
Axis in xy plane: in a plane perpendicular to the xy plane.
Axis perpendicular to xy plane: in the xy plane.

26. Reflection
Reflection about the x axis

27. Reflection
Reflection about the y axis

28. Reflection
Reflection relative to the coordinate origin

29. Reflection
Reflection of an object relative to an axis perpendicular to the xy plane through Prfl

30. Reflection
Reflection about the line y = x

31. Shear
The x-direction shear relative to x axis
If shx = 2:

32. Shear The x-direction shear relative to y = yref If shx = ½ yref = -1:
1/2
3/2

33. Shear The y-direction shear relative to x = xref If shy = ½ xref = -1:
3/2
If shy = ½ xref = -1:
1/2 1

34. Transformations between Coordinate Systemsx’ y’ θ y0 x’ θ x x x0

35. Transformations between Coordinate Systems
Method 1:
Method 2:

36. Affine Transformations
A coordinate transformation of the form
x’=axxx+axyy+bx, y’=ayxx+ayyy+by
x’ and y’ is a linear function of the original coordinates x and y.
aij and bk are constants determined by the transformation type.
Translation, rotation, scaling, reflection, and shear are two-dimensional affine transformations.
An affine transformation involving only rotation, translation, and reflection preserves angles and lengths.