Next: Solution of the minimization Up: Theory and implementation Previous: Optimal superposition. Contents

##### Quaternions and rotations.

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 Previous: Optimal superposition. Contents