Rodrigues Rotation Formula

Author

Eric W. Weisstein
March 12, 2009

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

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

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

Identity

MatrixForm[ωtilde = {{0,-(Subscript[ω, z]),(Subscript[ω, y])},{(Subscript[ω, z]),0,-(Subscript[ω, x])},{-(Subscript[ω, y]),(Subscript[ω, x]),0}}]//TraditionalForm

m1=FullSimplify[ComplexExpand[MatrixExp[θ ωtilde]/.(Subscript[ω, z])^2->1-(Subscript[ω, x])^2-(Subscript[ω, y])^2]]

m2=IdentityMatrix[3]+ωtilde Sin[θ]+ωtilde.ωtilde(1-Cos[θ])/.(Subscript[ω, z])^2->1-(Subscript[ω, x])^2-(Subscript[ω, y])^2

Simplify[m1-m2]

FullSimplify[m2/.-(Subscript[ω, x])^2-(Subscript[ω, y])^2->(Subscript[ω, z])^2-1]//MatrixForm//TraditionalForm

FullSimplify[ωtilde.{(Subscript[ω, x]),(Subscript[ω, y]),(Subscript[ω, z])}]