Transformations in 3 Dimensions CS /28/2018 Dr. Mark L. Hornick

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Presentation transcript:

Transformations in 3 Dimensions CS-321 11/28/2018 Dr. Mark L. Hornick

Pure translation CS-321 11/28/2018 Dr. Mark L. Hornick

3-D Rotation About an Axis CS-321 11/28/2018 3-D Rotation About an Axis y x z Positive rotation is counterclockwise, when looking from positive direction along an axis. CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Rotation About z-Axis Note similarity with the 2-D case CS-321 11/28/2018 Rotation About z-Axis Note similarity with the 2-D case CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Rotation About x-Axis CS-321 11/28/2018 Dr. Mark L. Hornick

CS-321 11/28/2018 Rotation About y-Axis Note transposition of the +/- on the sin() terms w.r.t. rotation about z and x CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

CS-321 11/28/2018 Compound 3-D Rotations Arbitrary orientations can be expressed as a result of successive rotations about each axis Z  Y  X The same orientation can also be expressed as a result of successive rotations about other axes X  Y  Z Z  Y  Z And 15 others… CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

CS-321 11/28/2018 Compound rotation Any arbitrary orientation can be represented with a given set of angles. CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

CS-321 Dr. Mark L. Hornick

Direction cosines CS-321 11/28/2018 Dr. Mark L. Hornick

Inverse rotation CS-321 11/28/2018 Dr. Mark L. Hornick

Inverse rotation CS-321 11/28/2018 Dr. Mark L. Hornick

General Transformation CS-321 11/28/2018 General Transformation CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

General Inverse Transformation CS-321 11/28/2018 General Inverse Transformation CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick

Scaling Matrix in 3 dimensions CS-321 11/28/2018 Scaling Matrix in 3 dimensions Note: Usually Sx = Sy =Sz CS-321 Dr. Mark L. Hornick Dr. Mark L. Hornick