1. 3D Transformation

2. Anil Verma, IOE Pulchowk
Transformation
A transformation is an operation that transforms or changes a shape .
There are several basic ways you can change a shape:
Translation (moving it)
Rotation (turning it round)
Scaling (making it bigger or smaller).
Shear (changing the main shape).
Reflection (mirroring the shape about axis).
Transforming an object means transforming all of its points
Anil Verma, IOE Pulchowk

3. Anil Verma, IOE Pulchowk
3D Transformation
Same as 2D.
Add z-axis and z-coordinate.
Use 4X4 homogenous matrix.
In part I, we discussed translation, rotation and scaling.
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4. Anil Verma, IOE Pulchowk
3D Translation
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5. 3D Scaling (relative to the origin point)
x, y and z values multiplied by scaling factors sx, sy and sz
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6. 3D Scaling (relative to the origin point)
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7. 3D Scaling (relative to fixed point)
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8. Anil Verma, IOE Pulchowk
3D Scaling (relative to fixed point)
Scaling with a Selected Fixed Position y y y y x x x z x z z z
Original position
Translate
Scaling
Inverse Translate
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3D Geometric Transformations

9. 3D Scaling (relative to fixed point)
for general scaling (relative to fixed point F)
where
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10. Anil Verma, IOE Pulchowk
3D Rotation
Coordinate-Axes Rotations
X-axis rotation
Y-axis rotation
Z-axis rotation
General 3D Rotations
Rotation about an axis that is parallel to one of the coordinate axes
Rotation about an arbitrary axis
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3D Geometric Transformations

11. Anil Verma, IOE Pulchowk
June 10, 2018
3D Rotation – (z-axis)
Rotating around the z axis:
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12. Anil Verma, IOE Pulchowk
June 10, 2018
3D Rotation – (x-axis)
Rotation around the X axis:
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13. Anil Verma, IOE Pulchowk
June 10, 2018
3D Rotation – (y-axis)
Rotation around the Y axis:
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14. Anil Verma, IOE Pulchowk
General 3D Rotations
Rotation 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
Anil Verma, IOE Pulchowk
3D Geometric Transformations

15. Anil Verma, IOE Pulchowk
General 3D Rotations
Rotation about an Arbitrary Axis
Basic Idea
Translate (x1, y1, z1) to the origin
Rotate (x’2, y’2, z’2) on to the z axis
Rotate the object around the z-axis
Rotate the axis to the original orientation
Translate the rotation axis to the original position y T
(x2,y2,z2) R
(x1,y1,z1)
R-1 x
T-1 z
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3D Geometric Transformations

16. Arbitrary Axis Rotation
Step 1. Translation
(x2,y2,z2)
(x1,y1,z1) x z y
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3D Geometric Transformations

17. Arbitrary Axis Rotation
Step 2. Establish [ TR ]x x axis
(a,b,c)
(0,b,c)
Projected Point 
Rotated Point x y z
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3D Geometric Transformations

18. Arbitrary Axis Rotation
Step 3. Rotate about y axis by 
(a,b,c)
(a,0,d)  l d x y
Projected Point z
Rotated Point
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3D Geometric Transformations

19. Arbitrary Axis Rotation
Step 4. Rotate about z axis by the desired angle  y l x  z
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3D Geometric Transformations

20. Arbitrary Axis Rotation
Step 5. Apply the reverse transformation to place the axis back in its initial position x y l z
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3D Geometric Transformations

21. Anil Verma, IOE Pulchowk
Example
Ex) Find the new coordinates of a unit cube 90º-rotated about an axis defined by its endpoints A(2,1,0) and B(3,3,1).
A Unit Cube
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3D Geometric Transformations

22. Anil Verma, IOE Pulchowk
Example
Step1. Translate point A to the origin x z y
B’(1,2,1)
A’(0,0,0)
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3D Geometric Transformations

23. Anil Verma, IOE Pulchowk
Example
Step 2. Rotate axis A’B’ about the x axis by and angle , until it lies on the xz plane. y
Projected point
(0,2,1)
B’(1,2,1) l  x z
B”(1,0,5)
Anil Verma, IOE Pulchowk
3D Geometric Transformations

24. Anil Verma, IOE Pulchowk
Example
Step 3. Rotate axis A’B’’ about the y axis by and angle , until it coincides with the z axis. y l  x
(0,0,6)
B”(1,0,  5) z
Anil Verma, IOE Pulchowk
3D Geometric Transformations

25. Anil Verma, IOE Pulchowk
Example
Step 4. Rotate the cube 90° about the z axis
Finally, the concatenated rotation matrix about the arbitrary
axis AB becomes,
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3D Geometric Transformations

26. Anil Verma, IOE Pulchowk
Example
Anil Verma, IOE Pulchowk
3D Geometric Transformations

27. Anil Verma, IOE Pulchowk
Example
Multiplying [TR]AB by the point matrix of the original cube
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3D Geometric Transformations

28. Anil Verma, IOE Pulchowk
3D Reflection
Reflection Relative to the xy Plane
[Do also Reflection Relative to the yz,zx Plane...] x z y
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3D Geometric Transformations

29. Anil Verma, IOE Pulchowk
3D Shear
Z-axis shear
Where a and b are the shear factors for x and y respectively.
Do, X-axis and Y-axis shear.
Anil Verma, IOE Pulchowk
3D Geometric Transformations