Euler Parameters

Author

Eric W. Weisstein
February 10, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/CoordinateGeometry/EulerParameters.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/EulerParameters.html.

©2006 Wolfram Research, Inc. except for portions noted otherwise

Definition

e0=Cos[ϕ/2];
e1=nx Sin[ϕ/2];
e2=ny Sin[ϕ/2];
e3=nz Sin[ϕ/2];
e={e1,e2,e3};

Identities

FullSimplify[Plus@@(e0^2+e.e)/.nz->(Sqrt[1-(nx^2)-(ny^2)])]

FullSimplify[{2e0^2-1,e0^2-e.e}/.nz->(Sqrt[1-(nx^2)-(ny^2)])]

FullSimplify[2 e e0]

e0=Cos[(1/2)(ϕ+ψ)]Cos[(1/2)θ];
e1=Cos[(1/2)(ϕ-ψ)]Sin[(1/2)θ];
e2=Sin[(1/2)(ϕ-ψ)]Sin[(1/2)θ];
e3=Sin[(1/2)(ϕ+ψ)]Cos[(1/2)θ];
e={e1,e2,e3};

FullSimplify[Plus@@(e0^2+e.e)]