1. FCC Graficación CAD/CAM 3D Transformations

2. Contents 1. Translation 2. Scaling 3. Rotation 4. Other Transformations

3. 1. Translation x y z TxTx TzTz TyTy (x, y, z)

4. 2. Scaling x y z (x, y, z) ∴

5. x y z x y z x y z x y z Relative Scaling

6. 3. Rotation 2D Rotation  r r  r r Rotation about z -axis is implicit !!!

7.  r r x y z ∴

8. By symmetry, x y z x y z

9. x y z u (x 1, y 1, z 1 ) (x 2, y 2, z 2 ) Rotation about an arbitrary axis

10.  Basic idea 1. Translate the object so that the rotation axis passes through the origin. 2. Rotate the object so that the rotation axis coincides with one of the coordinate axis. 3. Perform the specified rotation. 4. R -1 5. T -1 T R

12.  Step 1: Translation x y z u = (a, b, c) (x 1, y 1, z 1 ) (x 2, y 2, z 2 ) u

13.  Step 2: Aligning u with z -axis x y z u = (a, b, c)  (0, b, c) = u' (0, 0, 1) = u z x y z u'' = (a, 0, d) d a uxux

14. x y z d a uyuy (0, 0, 1) = u z  x y z ∴ ≡ 1 ≡ 1

15. ∴ In Step 2, for later use

16.  Step 3: Rotate about z -axis by a given angle x y z 

17.  Step 4: [R -1 ] = [R x (  ) -1 ][R y (  ) -1 ]  Step 5: [T -1 ] ∴ In summary,

18. 4. Concatenation  The successive application of a number of transformations can be achieved with a single transformation matrix, the concatenation of the sequence.  Suppose two transformations T 1 and T 2 are to be applied successively. The same effect can be achieved by the application of a single transformation T 3, which is simply the product of the matrices T 1 and T2. That is: The point (x, y, z) is transformed into (x’, y’, z’) by T 1 : [ x’ y’ z’ 1 ] = [ x y z 1 ] T 1 (1) The point (x’’, y’’, z’’) is generated by applying T2: [ x’’ y’’ z’’ 1 ] = [ x’ y’ z’ 1 ] T 2 (2) Substituting (1) in (2) gives: [ x’’ y’’ z’’ 1 ] = ([ x y z 1 ] T 1 )T 2 = [ x y z 1 ] (T 1 T 2 ) The order of application of the transformations must be preserved when the transformation matrices are multiplied together.