2D Transformation Homogenous Coordinates Scale/Rotate/Reflect/Shear: X’ = XT Translate: X’ = X + T Multiple values for the same point e.g., (2, 3, 6) and (4, 6, 12) are same points

2D Transformation Homogenous Coordinates x w y (x, y,w) (x/w, y/w,1) w=1

3D Transformation Scaling 4x4 matrix Non-rigid transformation Special value (-1) of scaling factor give reflection

3D Transformation Rotation About X-axis X Y Z

3D Transformation Rotation About Z-axis X Y Z

3D Transformation Rotation About Y-axis X Y Z

3D Transformation Rotations Orthogonal Matrix Rigid Transformation

3D Transformation Translation Rigid transformation

3D Transformation Shear Off diagonal elements Non-rigid transformation

3D Transformation Shear X Y  (x, y) (x’, y’)

3D Transformation Concatenation of Transformations Transformations: T 1, T 2, T 3 T1T1 T2T2 T3T3 X X’ Alternatively, X X’ T Pipeline unit T 1 T 2 T 3

3D Transformation Rotation about a fixed point Rotation of a cube about its center (about Z-axis) X Y Z C

3D Transformation Rotation about an arbitrary axis X Y Z O P Axis: P 0 (x 0, y 0, z 0 ), (C x, C y, C z ) Angle:  OP: Unit vector O: (x 0, y 0, z 0 ) Translation (-x 0, -y 0, -z 0 ) CxCx CzCz CyCy

3D Transformation Rotation about an arbitrary axis X Y Z O P Axis: P 0 (x 0, y 0, z 0 ), (C x, C y, C z ) Angle:  Rotation about X axis by  CxCx CzCz CyCy  d

3D Transformation Rotation about an arbitrary axis X Y Z O Axis: P 0 (x 0, y 0, z 0 ), (C x, C y, C z ) Angle:  Rotation about Y axis by  CxCx d d  1

3D Transformation Rotation about an arbitrary axis Complete Transformation

3D Transformation General

3D Viewing Projections A Projectors Projection Plane Center of Projection B B’ A’ Perspective

3D Viewing Projections Parallel Projectors Projection Plane At Infinity A’ B’ A B

3D Viewing Parallel Projections Orthographic Side View Front View Top View Z X Y

3D Viewing Parallel Projections Multiviews (x=0 or y=0 or z=0 planes), one View is not adequate True size and shape for lines On z=0 plane Orthographic

3D Viewing Parallel Projections Axonometric Additional rotation,translation and then projection on z=0 plane

3D Viewing Parallel Projections Three types Trimetric: No foreshortening is the same. Dimetric: Two foreshortenings are the same. Isometric: All foreshortenings are the same. Axonometric

3D Viewing Parallel Projections Trimetric Dimetric Isometric Axonometric

3D Viewing Parallel Projections Isometric Let there be 2 rotations a) about y-axis  b) about x-axis 

3D Viewing Parallel Projections Isometric Let there be 2 rotations a) about y-axis  b) about x-axis 

3D Viewing Parallel Projections Isometric

3D Viewing Parallel Projections Isometric

3D Viewing Parallel Projections Solving equation find ,  and f Isometric