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The rotational minimization problem can be elegantly solved by using quaternion algebra.
Quaternions are so-called hypercomplex numbers, having a real unit,
, and three imaginary units,
,
, and
. Since
(cyclic), quaternion multiplication is not commutative.
A possible matrix representation of an arbitrary quaternion,
 |
(4.53) |
reads
 |
(4.54) |
The components
are real numbers. Similarly as normal complex numbers allow one to represent rotations in a plane, quaternions
allow one to represent rotations in space. Consider the quaternion representation of a vector
, which is given by
 |
(4.55) |
and perform the operation
 |
(4.56) |
where
is a normalized quaternion,
 |
(4.57) |
The symbol
stands for `trace'. We note that a normalized quaternion is represented by an orthogonal
matrix.
may then be written as
 |
(4.58) |
where the components
, abbreviated as
, are given by
 |
(4.59) |
The matrix
is the rotation matrix defined in
(4.52).
Next: Solution of the minimization
Up: Theory and implementation
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pellegrini eric
2009-10-06