1. Computer Graphics: 3D Transformations
Kocaeli Universitesity
Computer Engineering Department
Fall 2011

2. Extending From 2d Approach
Methods for geometric transformations in three dimensions are extended from two-dimensional methods by including considerations for the z coordinate.
A three-dimensional position, expressed in homogeneous coordinates, is represented as a four-element column vector. Thus , each geometric transformation operator is now 4 by 4 matrix.

3. Translation

4. Rotation: z-axis rotation
3D Coordinate-Axis Rotations
z-axis rotation (counter-clockwise)

5. Rotation: x-axis rotation
counter-clockwise

6. Rotation: y-axis rotation
counter-clockwise

7. Scaling Change the coordinates of the object by scaling factors. y x z

8. Scaling with respect to a Fixed Point
Translate to origin, scale, translate back y x y y y x x x z z z z
Translate
Scale
Translate back

9. Scaling with respect to a Fixed Point

10. Reflection Reflection over planes, lines or points y y y y x x x x z z

11. Shear
Deform the shape depending on another dimension

12. Advanced Topics

13. 1. Rotating about an axis that is parallel to one of the coordinates 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 rotation axis is moved back to its original
A coordinate position P is transformed with the sequence

14. Rotation Around a Parallel Axis
Rotating the object around a line parallel to one of the axes: Translate to axis, rotate, translate back. y y y y x x x x z z z z
Translate
Rotate
Translate back

15. 2. Rotation Around an Arbitrary Axis
In this case, we also need rotation to align the rotation axis with a selected coordinate axis and then to bring the rotation axis back to its original orientation
A rotation axis can be defined with two coordinate position, or one position and direction angles.
Now we assume that the rotation axis is defined by two points, and that the direction of rotation is to be counter clockwise when looking along the axis from p2 to p1.

16. Rotation Around an Arbitrary Axis
Translate the object so that the rotation axis passes though the origin
Rotate the object so that the rotation axis is aligned with one of the coordinate axes
Make the specified rotation
Reverse the axis rotation
Translate back x z

17. Rotation Around an Arbitrary Axis

18. Rotation Around an Arbitrary Axis
u is the unit vector along V:
First step: Translate P1 to origin:
Next step: Align u with the z axis
we need two rotations: rotate around x axis to get u
onto the xz plane, rotate around y axis to get u aligned
with z axis.

19. Rotation Around an Arbitrary Axis
Align u with the z axis
1) rotate around x axis to get u into the xz plane,
2) rotate around y axis to get u aligned with the z axis y y y u u u' α α x x x u uz β z z z

20. Dot product and Cross Product
v dot u = vx * ux + vy * uy + vz * uz. That equals also to |v|*|u|*cos(a) if a is the angle between v and u vectors. Dot product is zero if vectors are perpendicular. v x u is a vector that is perpendicular to both vectors you multiply. Its length is |v|*|u|*sin(a), that is an area of parallelogram built on them. If v and u are parallel then the product is the null vector.

21. Rotation Around an Arbitrary Axis
Align u with the z axis
1) rotate around x axis to get u into the xz plane,
2) rotate around y axis to get u aligned with the z axis
We need cosine and sine of α for rotation u u' α x uz z
Projection of u on
yz plane

22. Rotation Around an Arbitrary Axis
Align u with the z axis
1) rotate around x axis to get u into the xz plane,
2) rotate around y axis to get u aligned with the z axis u β x
u''= (a,0,d) z