12. 3D Coordinate Operations ME 521 Computer Aided Design 12. 3D Coordinate Operations Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gmail.com Mechanical Engineering Department Gebze Technical University

Mechanical Engineering Department, GTU Introduction 12. 3D Coordinate Operations Most of the problems needs 3D computer graphics for descrition. For this purpose 3D Trasnformations or coordinate operations are used.   3-D Transformations: Translation Rotation Scaling Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU

Mechanical Engineering Department, GTU Translation 12. 3D Coordinate Operations Tx , Ty , Tz are the components of the translation in  x, y, z, in object coordinates Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU

Mechanical Engineering Department, GTU Rotation 12. 3D Coordinate Operations Through point (0, 0, 0) and about x axis, by the angle q counterclockwise when looked from +x: Below figure shows +X and –X rotation directions İnitial state -X rotation +X rotation (+y rotation direction) Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU

Mechanical Engineering Department, GTU Rotation 12. 3D Coordinate Operations Below figure shows +X +Y, and +Z rotation directions. +X rotation İnitial state +Z rotation +Y rotation Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU

Mechanical Engineering Department, GTU Rotation 12. 3D Coordinate Operations Transformation matrices for rotation: Rotation about X axis: Rotation about Y axis: Rotation about Z axis: Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU

Mechanical Engineering Department, GTU Scaling 12. 3D Coordinate Operations Transformation matrix for scaling: In below figure mirror images of two objects are onbtained by scaling. Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU

Mechanical Engineering Department, GTU Inverse Transforms 12. 3D Coordinate Operations Inverse transforms are obtained by placing negative values: Inverse of Translation: Inverse of Rotation abou  X: Inverse of Rotation about  Y: Inverse of Rotation about  Z: Inverse transform of T matrix is represented by T-1 . Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU

Mechanical Engineering Department, GTU Concatenation 12. 3D Coordinate Operations Concatenation of operations: or Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU

Mechanical Engineering Department, GTU Concatenation 12. 3D Coordinate Operations Performing the procedure must comply with the order of concatenation operations. Below example shows the difference between rotations about first +X and then +Y and the reverse. İnitial state +Y rotation +X rotation +Y rotation +X rotation İnitial state Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU