Euler Angles

Author

Eric W. Weisstein
January 21, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/EulerAngles.nb.

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Derivation

V5

<<Geometry`Rotations`

RotationMatrix3D[ϕ,θ,ψ]//TraditionalForm

V6

RotationMatrix3D[phi_,theta_,psi_]:=RotationMatrix[Pi-psi,{0,0,1}].RotationMatrix[theta,{1,0,0}].RotationMatrix[Pi-phi,{0,0,1}]

RotationMatrix3D[ϕ,θ,ψ]//TraditionalForm

{xp,yp,zp}=RotationMatrix3D[ϕ,θ,ψ].{x,y,z};

Delta[i_,j_]:=If[i==j,1,0];
Epsilon[i_,j_,k_]:=0;
Epsilon[1,2,3]:=1;
Epsilon[2,3,1]:=1;
Epsilon[3,1,2]:=1;
Epsilon[1,3,2]:=-1;
Epsilon[3,2,1]:=-1;
Epsilon[2,1,3]:=-1;
a[i_,j_]:=Delta[i,j](e[0]^2-Sum[e[k]e[k],{k,3}])+2e[i]e[j]+
  2Sum[Epsilon[i,j,k]e[0]e[k],{k,3}]

Array[a,{3,3}]/.e[n_]:>Subscript[e,n]//TraditionalForm

RotationMatrix3D[phi_,theta_,psi_]:=RotationMatrix[Pi-psi,{0,0,1}].RotationMatrix[theta,{1,0,0}].RotationMatrix[Pi-phi,{0,0,1}]

{
𝒟==RotationMatrix[-ϕ,{0,0,1}],
𝒞==RotationMatrix[θ,{1,0,0}],
ℬ==RotationMatrix[-ψ,{0,0,1}]
}//Column//TraditionalForm

{𝒟=RotationMatrix[-ϕ,{0,0,1}],
𝒞=RotationMatrix[θ,{1,0,0}],
ℬ=RotationMatrix[-ψ,{0,0,1}]};

ℬ.𝒞.𝒟//TraditionalForm

Relation to Cayley-Klein Parameters

Solve[{
alpha==E^(I(psi+phi)/2)Cos[theta/2],
 beta==I E^(I(psi-phi)/2)Sin[theta/2],
gamma==I E^(-I(psi-phi)/2)Sin[theta/2],
delta==E^(-(psi+phi)/2)Cos[theta/2]},{theta,psi,phi}]