1. 12. 3D Coordinate Operations
ME 521
Computer Aided Design
12. 3D Coordinate Operations
Assoc.Prof.Dr. Ahmet Zafer Şenalp
Mechanical Engineering Department
Gebze Technical University

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

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

10. 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