1. Geometric Transformation

2. So far…. We have been discussing the basic elements of geometric programming. We have discussed points, vectors and their operations and coordinate frames and how to change the representation of points and vectors from one frame to another. Next topic involves how to map points from one place to another (transformation).

3. Affine Transformation In this topic, we will concentrate on one particular transformation called affine transformations. Examples of affine transformations are: translations rotations Uniform and non-uniform scaling reflections (flipping objects about a line) shearing ( which deform squares into parallelogram)

4. Affine Transformation (cont.) Figure 1: Examples of affine transformation

5. Example Figure 2: Transformation in 2-D and 3-D

6. Characteristics These transformations all have a number of things in common. 1.They all map lines to lines ( parallel lines will still be parallel after the transformations) 1.Translation, rotation and reflection preserve the lengths of line segments and the angles between segments 3.Uniform scaling preserve angles but not length 4.Non-uniform scaling and shearing do not preserve angles or lengths

7. Usage Transformations are useful in a number of situations: 1. We can compose a scene out a number of objects

8. Usage (cont.)

9. 2. Can design a single “motif” and manipulate the motif to produce the whole shape of an object especially if the object has certain symmetries. Example of snowflake after reflections, rotations and translations of the motif

10. Usage (cont.) 3. To view an object from a different angle

11. Usage (cont.) 4. To produce animation

12. Affine Transformation The two special types of affine transformation are 1.Rigid transformation: These transformations preserve both angles and lengths (I.e. translations, rotations and reflections) 2. Orthogonal transformation: These transformations preserve angles but not necessarily length.

13. Affine Transformation (why?)  The most common transformations used in computer graphics.  They simplify operations to scale, rotate and reposition objects.  Several affine transformations can be combined into a simple overall affine transformation in terms of a compact matrix representation

14. Transformation Matrix (2-D) For a point P that map to Q after affine transformation has a matrix representation as shown: NB: the third of the resultant matrix will always be 1 for a point

15. Transformation Matrix (2-D) For a vector V that maps to W after affine transformation has the matrix representation as shown: NB: the third row of the resultant matrix will always be 0 for a vector

16. Transformation Matrix The scalar m that we have in the transformation matrix will determine which affine transformation that we want to perform.

17. OpenGL graphics pipeline Normally in OpenGL, many transformation processes occur before the final objects are displayed on the screen. Basically the object that we define in our world will go through the same Procedure.

18. Transformations Transformations: changes in size, shape and orientation that alter the coordinate descriptions of objects.  Usually, transformations are represented and calculated using matrices.  OpenGL also uses the same approach to perform transformations  Lets look in detail at each of the transformations listed earlier..

19. Sine and Cosine Sin = b/cif c = 1then b = sin Cos = a/cif c = 1then c = cos c a b Recall our algebra lesson!!

20. Sine and Cosine (cont.) 90 0180 270 y x z -

21. Sine and Cosine (cont.) Cos = x/zif z = 1then cos = x Sin = y/zif z = 1then sin = y Cos (- ) = x/z = cos Sin (- ) = -y/z = -sin Cos ( ) = cos cos - sin sin Sin ( ) = sin cos + cos sin

22. Matrices In graphics most of the time matrix operation that we will deal with is matrix multiplication. The formula for multiplication of matrix A with m x p dimension and matrix B with p x n dimension is: where

23. Example

24. Characteristics of Matrix Multiplication of matrices is not commutative AB != BA Multiplication of several matrices is associative A(BC) = (AB)C

25. Identity Matrix Set of matrices that when they multiply another matrix which reproduce that matrix is called identity matrices I.e: 1D 2D 3D 4D

26. Object Transformation vs Coordinate Transformation Object Transformation: alters the coordinates of each point on the object according to some rule. No change of coordinate system Coordinate Transformation: defines a new coordinate system in terms of the old one and then represents all of the object’s point in the new coordinate system

27. Translation  Reposition an object along a straight line path from one coordinate location to another  2D point is translated by adding translation distances to the x and y coordinates  When translating (x,y) to (x’,y’) by value t x’ = x + t x y’ = y + t y

28. Translation (cont.) The (t x,t y ) is the translation distances called TRANSLATION VECTOR In matrix form Then point P’ = P + T

29. Translation (cont.) For 2-D translation the transformation matrix T has the following form: Where m x and m y are the translation values in x and y axis T =

30. Translation (cont.) For 3-D translation the transformation matrix T has the following form: Where m x, m y and m z are the translation values in x, y and z axis T =

31. Translation (cont.) Translation is a rigid body transformation the object is not being deformed all points are moved in the same distance How to translate? straight lines – based on end points polygons - based on vertices Circles - based on centre

32. Rotations  Reposition an object along a circular path in a specified Plane  Rotations are specified by: a rotation angle a rotation point (pivot point) at position (x,y)  Positive rotation angle means rotate counter-clockwise negative rotation angle means rotate clockwise

33. Rotations (cont.) (x,y) (x’,y’) r r The original point (x,y) can be represented in polar coordinates form: x = r cos(1) y = r sin (2)

34. Rotations (cont.) The new point (x’,y’) can be expressed as x’ = r cos ( ) y’ = r sin ( ) These equation can be written as x’ = r cos cos - r sin sin y’ = r cos sin + r sin cos

35. Rotations (cont.) Substituting the equations with equation (1) and (2), we get x’ = x cos - y sin y’ = x sin + y cos Therefore, the rotation matrix R can be expressed as R =

36. Rotations (cont.) For 2-D rotation the transformation matrix R has the following Form (in homogeneous coordinate system): NB: for counter-clockwise rotation

37. Rotations (cont.) In 3-D world, the rotation is more complex since we have to Consider rotation about three different axis; x, y and z axis. Therefore we have three different transformation matrices in 3-D world. R z ( ) = Rotation about z-axis (ccw)

38. Rotations (cont.) R x ( ) = R y ( ) = Rotation about y-axis (ccw) Rotation about x-axis (ccw)

39. Rotations (cont.) This rotation matrix is for case where the rotation is at the origin. For a rotation at any other points, we need to perform the following transformations: 1. Translating the object so the rotation point is at the origin. 2. Rotating the object around the origin 3. Translating the object back to its original position Recall that rotation is a rigid affine transformation

40. Scaling  Scaling changes the size of an object  Scaling can also reposition the object but not always  Uniform scaling is performed relative to some central fixed point (I.e at the origin). The scaling value (scale factor) for uniform scaling must be both equal.  Non-uniform scaling has different scaling factors. Also refers as differential scaling

41. Scaling (cont.) The value of the scale factors (S x, S y, S z ) determine the size of the scaled object. if equal to 1 -> no changes if greater than 1 -> increase in size (magnification) if 0 decrease in size (demagnification) if negative value -> reflection !!!

42. Scaling (cont.) For 2-D scaling the transformation matrix S will have the following form S =

43. Scaling (cont.) For 3-D scaling the transformation matrix S has the following form: S =