1. Review: 3D Transforms Continued
Rotation about an arbitrary axis
Quaternions

2. 3D rotations about an x,y, or z axis are trivial extensions of 2D rotations about the origin
For example, consider a rotation of 90 degrees about the x-axis. The y and z coordinates change but the x coordinate is not affected.
A 3D rotation about one of the major axes occurs in a 2D plane defined by the point’s coordinate relative to that axis (x = 0 plane in this case).
(0,d,0) Z
(0,0,d) X
©Babu 2009
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3. 3D Rotation about the x axis
By looking at the 2D rotation matrix we can write down what the 3D rotation matrix about the x axis must look like:
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©Babu 2009

4. Translations are defined relative to the x,y, and z axes
Notation: P’ = Tx(5)P
Implementation
Meaning: Move the position of point P five units in a positive direction with respect to the x axis
New point is (x+5, y, z)
©Babu 2009
5/14/2018

5. Rotation about the x axis
Notation P’ = Rx(30)P
Implementation (Homogenous Coordinates)
Meaning
Rotate a point P a positive degrees about the x-axis
©Babu 2009
5/14/2018

6. Rotation about the y axis
Notation P’ = Ry(30)P
Implementation (Homogenous Coordinates)
Meaning
Rotate a point P a positive degrees about the y-axis z x y
©Babu 2009
5/14/2018

7. Rotation about the z axis
Notation P’ = Rz(30)P
Implementation (Homogenous Coordinates)
Meaning
Rotate a point P a positive degrees about the z-axis
©Babu 2009
5/14/2018

8. Rotation About An Arbitrary Axisp2 u Z θ p1 Y
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©S. Babu X

9. Rotation About An Arbitrary Axis
Translate p1 to origin p2
T(-p1) u Z p1 u' Y
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©S. Babu X

10. Rotation About An Arbitrary Axis
Rotate u' to u'' in YZ plane (rotation about Y axis, Ry(-β))
Ry(-β)) ∙ T(-p1) Z u' uz c Y a ß uy b ux
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©Z. Wartell, S. Babu, L. F. Hodges X

11. Rotation About An Arbitrary Axis
Rotate u' to u'' in YZ plane (rotation about Y axis, Ry(-β))
Ry(-β)) ∙ T(-p1) Z
u'' u' uz c Y a ß uy b ux
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©Z. Wartell, S. Babu, L. F. Hodges X

12. Rotation About An Arbitrary Axis
After Ry(-ß), angle α and u'' lies in the y-z plane.
Next, Rotate u'' to Z axis (u''') (rotation about X axis, Rx(α))
Rx(α) ∙ Ry(-β) ∙ T(-p1)
u'' Z a c uz Y α uy ux
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©S. Babu X

13. Rotation About An Arbitrary Axis
After Rx(α), u'' lies in the Z axis as u'''.
Next, rotate about Z by θ (Rz(θ))
Rz(θ) ∙ Rx(α) ∙ Ry(-β) ∙ T(-p1)
u''' Z
u'' Y θ
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©Z. Wartell, S. Babu, L. F. Hodges X

14. Rotation About An Arbitrary Axis
Now we’ve rotated about U.
Next, apply the inverse transformations to place u''' back on u p2
R((p1,p2),θ) = T(p1) ∙ Ry(β) ∙ Rx(-α) ∙ Rz(θ) ∙ Rx(α) ∙ Ry(-β) ∙ T(-p1) u Z θ p1 Y
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©Z. Wartell, S. Babu, L. F. Hodges X

15. Rotation About An Arbitrary Axis
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©Z. Wartell, S. Babu

16. Quaternions Alternative rotation representation: 4 real numbers
interpolation between quaternions well defined
(“slerp”)
no gimbal lock
composition of quaternions: 16 *’s, 9+’s
(matrix 27 *’s and 18 +’s)
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©Z. Wartell, S. Babu

17. Quaternions Rotation representation:
Relation to axis-angle representation (u,θ) is: y u θ x Z
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©S. Babu

18. Quaternion Rotation: Basic Idea
To rotate point p:
represent p as quaternion:
compute rotated quaternion point representation
extract standard coordinate representation p' from qp' y u p' p x Z
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©S. Babu

19. Details of Rotation How do you multiply quaternions:
How do you find inverse of quaternion:
For rotations we assume unit quaternions, hence:
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©S. Babu

20. Quaternion Rotation: Complete Equation
To rotate point p: y u p' p x Z
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©S. Babu

21. How do quaternions help?
Quaternion composition: requires 16 *’s, 12+’s while 3x3 matrix composition needs 27*’s and 18+’s
No gimbal lock
interpolation between two orientations using “spherical linear interpolation” (slerp) of their quaternions: -constant angular velocity
-unique interpolation
-interpolation that is same regardless of coordinate
system used for computation
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©Z. Wartell, S. Babu