Page 6 - Computer Graphics Handout
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3.6.5 Interpolation ................................................................................................................................................... 112
3.6.6 Displaying the Cube ........................................................................................................................................ 112
3.7 AFFINE TRANSFORMATIONS .................................................................................................................................. 113
3.8 TRANSLATION, ROTATION, AND SCALING ............................................................................................................. 115
3.8.1 Translation ...................................................................................................................................................... 115
3.8.2 Rotation .......................................................................................................................................................... 115
3.8.3 Scaling ............................................................................................................................................................. 117
3.9 TRANSFORMATIONS IN HOMOGENEOUS COORDINATES ..................................................................................... 118
3.9.2 Scaling ............................................................................................................................................................. 119
3.9.3 Rotation .......................................................................................................................................................... 119
3.9.4 Shear ............................................................................................................................................................... 121
3.10 CONCATENATION OF TRANSFORMATIONS ......................................................................................................... 122
3.10.1 Rotation About a Fixed Point ........................................................................................................................ 122
3.10.2 General Rotation ........................................................................................................................................... 124
3.10.3 The Instance Transformation ........................................................................................................................ 125
3.10.4 Rotation About an Arbitrary Axis .................................................................................................................. 126
3.11 TRANSFORMATION MATRICES IN OPENGL.......................................................................................................... 128
3.11.1 Current Transformation Matrices ................................................................................................................. 129
3.11.2 Rotation, Translation, and Scaling ................................................................................................................ 129
3.11.4 Order of Transformations ............................................................................................................................. 131
3.12 SPINNING OF THE CUBE ....................................................................................................................................... 131
3.12.1 Updating in the Display Callback .................................................................................................................. 131
3.12.2 Uniform Variables ......................................................................................................................................... 132
3.13 INTERFACES TO THREE-DIMENSIONAL APPLICATIONS ........................................................................................ 133
3.13.1 Using Areas of the Screen ............................................................................................................................. 133
3.13.2 A Virtual Trackball ......................................................................................................................................... 134
3.13.3 Smooth Rotations ......................................................................................................................................... 135
3.13.4 Incremental Rotation .................................................................................................................................... 136
3.14 QUATERNIONS ..................................................................................................................................................... 137
3.14.1 Complex Numbers and Quaternions ............................................................................................................ 137
3.14.2 Quaternions and Rotation ............................................................................................................................ 138
3.15 Curve Generation and Interpolation ................................................................................................................... 139
3.15.1 Parametric Curves ......................................................................................................................................... 140
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