Applications 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.

quaternions

it follows euler s theorem relative orientation of pair of coordinate systems may specified set of 3 independent numbers. redundant fourth number added simplify operations quaternion algebra. 3 of these numbers direction cosines orient eigenvector. fourth angle eigenvector separates 2 sets of coordinates. such set of 4 numbers called quaternion.

while quaternion described above, not involve complex numbers, if quaternions used describe 2 successive rotations, must combined using non-commutative quaternion algebra derived william rowan hamilton through use of imaginary numbers.

rotation calculation via quaternions has come replace use of direction cosines in aerospace applications through reduction of required calculations, , ability minimize round-off errors. also, in computer graphics ability perform spherical interpolation between quaternions relative ease of value.