1. On Cayley’s Factorization of 4D Rotations and Applications
Federico Thomas and Alba Pérez-Gracia
IRI (CSIC-UPC) and Idaho State University

2. Preamble
In Computational Kinematics, dual quaternions are used to model the movement of solid objects in 3D, i.e., to represent the group of spatial displacements SE(3).
Dual Quaternion
G+ 0,3,1 1 i
e23 j
e31 k
e12 iε
e41 jε
e42 kε
e43 ε
e1234

3. Outline of the presentation
Rotations in 4D
Isoclinic rotations
Cayley’s factorization
Elfrinkhof-Rosen method
Our alternative
Rotations in 3D
A useful mapping
Example
Conclusion

4. Rotations in 4D
By properly choosing the reference frame, a rotation in 4D can be expressed as:

5. Isoclinic rotations
Cayley’s factorization: any 4D rotation matrix can be decomposed into the product of a right- and a left-isoclinic matrix.
The product of a right- and a left-isoclinic matrix is commutative.
The product of two right- (left-) isoclinic matrices is a right- (left-) isoclinic matrix.

6. Isoclinic rotations

7. Isoclinic rotations
A rotation in 4D can be represented as a double quaternion

8. Cayley’s factorization (Elfrinkhof-Rosen method)
If we square and add all the elements in row i

9. Cayley’s factorization (our method)

10. Cayley’s factorization (our method)
We can define the following linear operators:

11. Rotations in 3D

12. Rotations in 3D

13. Rotations in 3D

14. Rotations in 3D
The representation based on a double quaternion is redundant.
One quaternion is enough to represent a 3D rotation

15. From quaternion to matrix representation
A rotation in 3D can be either represented as or
How to recover the matrix representation?
Easy!

16. Geometric interpretation of
Alternative expression to Rodrigues’ formula

17. A useful mapping We define the “magic” mapping:
The interest of this mapping is that, applying Cayley’s factorization, we get

18. Geometric interpretation
Chasles’ theorem d

19. Example

20. Example

21. Conclusion
Cayley’s factorization is a fundamental tool in Computational Kinematics. It can be seen as a unifying procedure to:
obtain the double quaternion representation of 4D rotations,
the dual quaternion representation of 3D displacements (as a particular case, the quaternion representation of 3D rotations).

22. Degenerate cases >> [L R]= CayleyFactorization(trotz(0)) L =
L =
= I
R = 
>> [L R]= CayleyFactorization(trotz(pi))
L =
= A_3
R = 
= B_3