Euler Angles

According to Euler's rotation theorem, any rotation may be described using three angles. If the rotations are written in terms of rotation matrices B, C, and D, then a general rotation can be written as A=DCB.

Rotations around the z�x�z axes.

cos sin 0

sin cos 0

0 0 1

B

φ φφ φ

− =

1 0 0

0 cos sin

0 sin cos

C θ θθ θ

= −

cos sin 0

sin cos 0

0 0 1

D

ψ ψψ ψ

− =

cos( ) sin( ) sin( ) cos( )

cos( )

cos( ) cos( ) sin( )

cos(

- - -

+ -

cos( ) sin( ) cos( ) sin( ) sin( )

sin( ) cos( ) sin( ) cos( )) cos( ) sin( )

sin( ) sin( ) cos

sin( ) sin( ) cos( )

sin( ) cos( )

o (- )+ c s

CBDA

θ θ θθ θ θ

θ

φ φ φ φφ φ φ φ

φ φ

ψ ψ

θ

ψ ψ ψψ ψ ψ ψ ψ

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅

=

=( )θ

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