1. Computer Graphics University of Palestine Dr. Sana’a Wafa Al-Sayegh
ITGD3107
Prepared By
Niddal abu swereh
Mahmoud elqedra
Supervised By
Dr. Sana’a Wafa Al-Sayegh

2. ITGD3107 Computer Graphics
Chapter 11
Three-Dimensional Geometric and Modeling Transformations

3. Three-Dimensional Geometric and Modeling Transformations
Some Basics
3D Translations.
3D Scaling.
3D Rotation.
3D Reflections.
Transformations.

4. Some Basics Basic geometric types. Scalars s Vectors v Points p
Transformations
Types of transformation:
rotation, translation, scale, Reflections, shears.
Matrix representation
Order
P=T(P)

5. 3D Point
We will consider points as column vectors. Thus, a typical point with coordinates (x, y, z) is represented as:

6. 3D Translations. P is translated to P' by T: T = Called the
matrix
T =

7. 3D Translations.

8. 3D Translations.
An object is translated in 3D dimensional by transforming each of the defining points of the objects.

9. 3D Translations.

10. 3D Scaling
P is scaled to P' by S:
Called the
Scaling matrix
S =

11. 3D Scaling
Scaling with respect to the coordinate origin

12. 3D Scaling
Scaling with respect to a selected fixed position (xf, yf, zf)
Translate the fixed point to origin
Scale the object relative to the coordinate origin
Translate the fixed point back to its original position

13. 3D Scaling

14. About an axis: equivalent to 180˚rotation about that axis
3D Reflections
About an axis: equivalent to 180˚rotation about that axis

15. 3D Reflections

16. 3D Shearing Modify object shapes Useful for perspective projections:
E.g. draw a cube (3D) on a screen (2D)
Alter the values for x and y by an amount proportional to the distance from zref

17. 3D Shearing

18. Shears

19. Rotation
Positive rotation angles produce counterclockwise rotations about a coordinate axis
&&

20. Rotation
&&

21. Coordinate-Axes Rotations
&&

22. Coordinate-Axes Rotations
&&

23. Coordinate-Axes Rotations
&&

24. Coordinate-Axes Rotations
&&

25. General Three-Dimensional Rotations
An object is to be rotated about an axis that is parallel to one of the coordinate axes
Translate the object so that the rotation axis coincides with the parallel coordinate axis
Perform the specified rotation about that axis
Translate the object so that the rotation axis is moved back to its original position
&&

26. General Three-Dimensional Rotations
An object is to be rotated about an axis that is not parallel to one of the coordinate axes
Translate the object so that the rotation axis passes through the coordinate origin.
Rotate the object so that the axis of rotation coincide with one of the coordinate axes.
Perform the specified rotation about that coordinate axis.
Apply inverse rotations to bring the rotation axis back to its original orientation.
Apply the inverse Translation to bring the rotation axis back to its original position.
&&

27. Quiz
Draw any shape, then moving translation matrix.
Good Luck