1. DEPT. OF APPLIED MATHEMATICS UNIVERSITY OF STELLENBOSCH
THE 3x3 ROTATION MATRIX
A geometrical derivation
MILTON F. MARITZ
DEPT. OF APPLIED MATHEMATICS
UNIVERSITY OF STELLENBOSCH
SANUM, 31st March 2003

2. 3x3 Rotation matrices on the principal axes  Rotation on x-axis
 Rotation on y-axis
 Rotation on z-axis

3. The general 3x3 rotation matrix
Q(a,q) a b’ b

4. The general 3x3 rotation matrix
Q(a,q) a b’ b

5. Anton & Rorres
Strang

6. The general 3x3 rotation matrix
Q(a,q)
FORWARD PROBLEM:
Given a and q, find Q
(2) INVERSE PROBLEM:
Given Q, find a and q

7. Eigenvalue decomposition
Decompose Q as
Then, ,
where a is the (normalised) axis vector,
and q is the rotation angle.

8. Projections are needed

9. Projections on and perpendicular to a
a is normalized a

10. Projections on and perpendicular to a
a is normalized a b

11. Projections on and perpendicular to a
a is normalized a q b

12. Projections on and perpendicular to a
a is normalized a q b p

13. The cross product is also needed

14. The cross product

15. The cross product The cross product, , can be expressed as
matrix multiplication.
For a given , define the cp-matrix
Then

16. The cp matrix is anti symmetric
Some useful properties of A (for a normalized):
is anti symmetric
a is an eigenvector

17. The square of the cp matrix
Some useful properties of (for a normalized):
---it is a projection matrix,
projection onto a plane
is symmetric

18. The cp matrix
Some useful properties of (for a normalised):

19. The cp matrix
Some useful properties of (for a normalised):

20. The cp matrix
Some useful properties of (for a normalised):

21. The cp matrix
Some useful properties of (for a normalised):

22. The cp matrix
Some useful properties of (for a normalised):

23. A geometrical derivation
3x3 Rotation matrix
A geometrical derivation a b

24. A geometrical derivation
3x3 Rotation matrix
A geometrical derivation a b’ b

25. A geometrical derivation
3x3 Rotation matrix
A geometrical derivation a
p=(I- aaT)b
=-A2 b b’ b p’ p

26. A geometrical derivation
3x3 Rotation matrix
A geometrical derivation a
p=(I- aaT)b
=-A2 b
q = aaT b
= (I+A2)b b’ q b p’ p

27. A geometrical derivation
3x3 Rotation matrix
A geometrical derivation a
p=(I- aaT)b=A2 b
p=(I- aaT)b
=-A2 b
q = aaT b
= (I+A2)b
c = A b b’ q b c p’ p

28. A geometrical derivation
3x3 Rotation matrix
A geometrical derivation
p=(I- aaT)b=A2 b
p=(I- aaT)b
=-A2 b
q = aaT b
= (I+A2)b
c = A b
p’=p cq + c sq c p’
c sin q
p cos q p

29. A geometrical derivation
3x3 Rotation matrix
A geometrical derivation
p=(I- aaT)b
=-A2 b
q = aaT b
= (I+A2)b
c = A b
p’=p cq + c sq
b’=q + p’ b’ q p’

30. A geometrical derivation
3x3 Rotation matrix
A geometrical derivation
c = A b c in terms of A and b
p=(I- aaT)b=-A2 b
p’=p cq + c sq
= -cq A2 b + sq A b
p’ in terms of A,q and b
q = aaT b
= (I+A2)b
q in terms of A and b
b’= q + p’
=[I + A sq + (1- cq )A2 ]b
=Q b where Q = I + A sq + (1- cq )A2
p=(I- aaT)b
=-A2 b
q = aaT b
= (I+A2)b
c = A b
p’=p cq + c sq
b’=q + p’

31. Solution to the FORWARD PROBLEM
Q = I + A sq + (1- cq )A2
where and
Given a and q, find Q

32. Solution to the
INVERSE PROBLEM
Given Q, find a and q:
Tr(Q) =Tr(I) + Tr(A) sq + (1- cq )Tr (A2)
= (1- cq )(-2)
= 1 + 2cq q found
Q = I + A sq + (1- cq )A2
QT= I - A sq + (1- cq )A2
Q-QT = 2sq A A found

33. Some Properties
to be proved
QTQ=I
Q(a,-q)= QT(a,q)
Q(a,q)Q(a, f)=Q(a, q+f)
Q=eqA compact formula

34. Some Applications

35. Some Applications

36. Thank you for your attention !