Generators of rotations Euler's rotation theorem

suppose specify axis of rotation unit vector [x, y, z], , suppose have infinitely small rotation of angle Δθ vector. expanding rotation matrix infinite addition, , taking first order approach, rotation matrix Δr represented as:

Δ
r
=


[



1


0


0




0


1


0




0


0


1



]


+


[



0


z


−
y




−
z


0


x




y


−
x


0



]



Δ
θ
=

i

+

a


Δ
θ
.


{\displaystyle \delta r={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}+{\begin{bmatrix}0&z&-y\\-z&0&x\\y&-x&0\end{bmatrix}}\,\delta \theta =\mathbf {i} +\mathbf {a} \,\delta \theta .}

a finite rotation through angle θ axis may seen succession of small rotations same axis. approximating Δθ θ/n n large number, rotation of θ axis may represented as:

r
=


(

1

+




a

θ

n


)


n


≈

e


a

θ


.


{\displaystyle r=\left(\mathbf {1} +{\frac {\mathbf {a} \theta }{n}}\right)^{n}\approx e^{\mathbf {a} \theta }.}

it can seen euler s theorem states rotations may represented in form. product aθ generator of particular rotation, being vector (x,y,z) associated matrix a. shows rotation matrix , axis–angle format related exponential function.

one can derive simple expression generator g. 1 starts arbitrary plane (in euclidean space) defined pair of perpendicular unit vectors , b. in plane 1 can choose arbitrary vector x perpendicular y. 1 solves y in terms of x , substituting expression rotation in plane yields rotation matrix r includes generator g = ba − ab.

x




=

a

cos
⁡
α
+

b

sin
⁡
α





y




=
−

a

sin
⁡
α
+

b

cos
⁡
α




cos
⁡
α



=


a



t




x





sin
⁡
α



=


b



t




x






y




=
−


a
b



t




x

+


b
a



t




x

=

(


b
a



t



−


a
b



t



)


x







x

′




=

x

cos
⁡
β
+

y

sin
⁡
β






=

(

i

cos
⁡
β
+

(


b
a



t



−


a
b



t



)

sin
⁡
β
)


x






r




=

i

cos
⁡
β
+

(


b
a



t



−


a
b



t



)

sin
⁡
β






=

i

cos
⁡
β
+

g

sin
⁡
β





g




=


b
a



t



−


a
b



t









{\displaystyle {\begin{aligned}\mathbf {x} &=\mathbf {a} \cos \alpha +\mathbf {b} \sin \alpha \\\mathbf {y} &=-\mathbf {a} \sin \alpha +\mathbf {b} \cos \alpha \\\cos \alpha &=\mathbf {a} ^{\mathsf {t}}\mathbf {x} \\\sin \alpha &=\mathbf {b} ^{\mathsf {t}}\mathbf {x} \\[8px]\mathbf {y} &=-\mathbf {ab} ^{\mathsf {t}}\mathbf {x} +\mathbf {ba} ^{\mathsf {t}}\mathbf {x} =\left(\mathbf {ba} ^{\mathsf {t}}-\mathbf {ab} ^{\mathsf {t}}\right)\mathbf {x} \\[8px]\mathbf {x} &=\mathbf {x} \cos \beta +\mathbf {y} \sin \beta \\&=\left(\mathbf {i} \cos \beta +\left(\mathbf {ba} ^{\mathsf {t}}-\mathbf {ab} ^{\mathsf {t}}\right)\sin \beta \right)\mathbf {x} \\[8px]\mathbf {r} &=\mathbf {i} \cos \beta +\left(\mathbf {ba} ^{\mathsf {t}}-\mathbf {ab} ^{\mathsf {t}}\right)\sin \beta \\&=\mathbf {i} \cos \beta +\mathbf {g} \sin \beta \\[8px]\mathbf {g} &=\mathbf {ba} ^{\mathsf {t}}-\mathbf {ab} ^{\mathsf {t}}\end{aligned}}}

to include vectors outside plane in rotation 1 needs modify above expression r including 2 projection operators partition space. modified rotation matrix can rewritten exponential function.

p

a
b






=
−


g


2







r




=

i

−


p

a
b



+

(

i

cos
⁡
β
+

g

sin
⁡
β
)



p

a
b



=

e


g

β








{\displaystyle {\begin{aligned}\mathbf {p_{ab}} &=-\mathbf {g} ^{2}\\\mathbf {r} &=\mathbf {i} -\mathbf {p_{ab}} +\left(\mathbf {i} \cos \beta +\mathbf {g} \sin \beta \right)\mathbf {p_{ab}} =e^{\mathbf {g} \beta }\end{aligned}}}

analysis easier in terms of these generators, rather full rotation matrix. analysis in terms of generators known lie algebra of rotation group.