Quaternions Euler's rotation theorem

it follows euler s theorem relative orientation of pair of coordinate systems may specified set of 3 independent numbers. redundant fourth number added simplify operations quaternion algebra. 3 of these numbers direction cosines orient eigenvector. fourth angle eigenvector separates 2 sets of coordinates. such set of 4 numbers called quaternion.

while quaternion described above, not involve complex numbers, if quaternions used describe 2 successive rotations, must combined using non-commutative quaternion algebra derived william rowan hamilton through use of imaginary numbers.

rotation calculation via quaternions has come replace use of direction cosines in aerospace applications through reduction of required calculations, , ability minimize round-off errors. also, in computer graphics ability perform spherical interpolation between quaternions relative ease of value.