1. 10. 3D Coordinate Operations e-mail: Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gyte.edu.trazsenalp@gyte.edu.tr Makine Mühendisliği Bölümü Gebze Yüksek Teknoloji Enstitüsü ME 521 Computer Aided Design

2. Introduction Most of the problems needs 3D computer graphics for descrition. For this purpose 3D Trasnformations or coordinate ıperations are used. 3-D Transformations:  Translation  Rotation  Scaling Dr. Ahmet Zafer Şenalp ME 521 2 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

3. T x, T y, T z are the components of the translation in x, y, z, in object coordinates Dr. Ahmet Zafer Şenalp ME 521 3 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

4. Through point (0, 0, 0) and about x axis, by the angle  clockwise when looked from +x: Below figure shows +X and –X rotation directions (+y rotation direction) Dr. Ahmet Zafer Şenalp ME 521 4 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations İnitial state +X rotation -X rotation

5. Below figure shows +X +Y, and +Z rotation directions. Dr. Ahmet Zafer Şenalp ME 521 5 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations İnitial state +X rotation +Y rotation +Z rotation

6. Transformation matrices for rotation: Rotation about X axis: Rotation about Y axis: Rotation about Z axis: Dr. Ahmet Zafer Şenalp ME 521 6 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

7. Transformation matrix for scaling: In below figure mirror images of two objects are onbtained by scaling. Dr. Ahmet Zafer Şenalp ME 521 7 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

8. 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 8 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

9. or Concatenation of operations: Dr. Ahmet Zafer Şenalp ME 521 9 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

10. 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. Dr. Ahmet Zafer Şenalp ME 521 10 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations İnitial state +X rotation İnitial state +Y rotation +X rotation