The document discusses 2D and 3D transformations used in computer graphics. It covers topics like scaling, rotation, translation and how they can be represented using matrices. Matrix representations allow multiple transformations to be combined through matrix multiplication. Both linear transformations like scaling and rotation, as well as affine transformations like translation, can be captured with matrices. The order of matrix multiplications is important, as transformations are not commutative. These concepts extend from 2D to 3D graphics using homogeneous coordinates.

4. 2D Modeling Transformations Scale Rotate Translate Scale Translate x y World Coordinates Modeling Coordinates

8. 2-D Rotation x’ = x cos (  ) - y sin (  ) y’ = x sin (  ) + y cos (  )  (x, y) (x’, y’)

9. 2-D Rotation x = r cos (  ) y = r sin (  ) x’ = r cos (  +  ) y’ = r sin (  +  ) Trig Identity… x’ = r cos(  ) cos(  ) – r sin(  ) sin(  ) y’ = r sin(  ) sin(  ) + r cos(  ) cos(  ) Substitute… x’ = x cos (  ) - y sin (  ) y’ = x sin (  ) + y cos (  )   (x, y) (x’, y’)

40. Matrix Composition Will this sequence of operations work?

44. Basic 3D Transformations Identity Scale Translation Mirror about Y/Z plane

45. Basic 3D Transformations Rotate around Z axis: Rotate around Y axis: Rotate around X axis: