1. Two-Dimensional Geometric Transformations A two dimensional transformation is any operation on a point in space (x, y) that maps that point's coordinates into a new set of coordinates (x’, y’). Instead of applying a transformation to every point in every line that makes up an object, the transformation is applied only to the vertices of the object and then new lines are drawn between the resulting endpoints.

2. Two-Dimensional Geometric Transformations Basic Transformations Translation Rotation Scaling Composite Transformations Other transformations Reflection Shear Transformation include change in size, shape & orientation

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

4. Matrix Representation Translation matrix Row vector representation P = (x, y)‏T = (tx ty)‏P = (x, y)‏ P’= (x’, y’)‏P’ = P + T x ‘ = x + tx y’ = y + ty

5. Rotation Repositioning along a circular path in the xy plane y xr yr ө P P’ (xr, yr) is the rotation pointө is the rotation angle

6. Rotation  Rotation transformation x’=rcos(φ+θ)= rcos φ cos θ -rsin φ sin θ y’=rsin(φ + θ)= rcos φ sin θ +rsin φ cos θ x=rcos φ y=rsin φ P’= R· P x’ = x cos ө - y sin ө y’ = x sin ө + y Cos ө r r Φ ө P’ P (x’, y’)‏ (x, y)‏

7. (x’,y’)‏ (xr,yr)‏ ө Φ Rotation about arbitrary pivot point position x’ = xr + (x - xr) cosө - (y – yr) sin ө y’ = yr + (x - xr) sin ө - (y – yr) cos ө

8. General Pivot-Point Rotation Rotations about any selected pivot point (x r,y r ) Translate-rotate-translate

9. General Pivot Point Rotation

10. To rotate about P1 (x1,y1), the following sequence of the fundamental transformations are needed: 1.Translate the object by (-x1, -y1) 2.Rotate (Ø) 3.Translate the object by (x1, y1)

11. General Pivot-Point Rotation

12. Scaling  Alters the size of an object  Scaling an object is implemented by scaling the X and Y coordinates of each vertex in the object.

13. Scaling  Scaling transformation Scaling factors, s x and s y Uniform scaling,if s x = s y x y x y

14. Matrix Representations and Homogeneous Coordinates - allow all transformations as matrix multiplications Homogeneous Coordinates Matrix representations Translation h =1 for 2 D transformations

15. Matrix Representations  Matrix representations Scaling Rotation

16. Composite Transformations - sequence of transformations  Translations

17. Composite Transformations  Scaling

18. Composite Transformations  Rotations

19. General Scaling Directions  Scaling factors s x and s y scale objects along the x and y directions.  We scale an object in other directions with scaling factors s 1 and s 2

20. General Scaling Directions

21. 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 scaling rotation and uniform scaling

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

23. 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 180 0 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.