Appendix/Mathematics/EulerAngles

Euler Angles

Here we will follow quite closely the online class notes prepared by Professor Mona Berciu, Department of Physics & Astronomy, University of British Columbia.

Rotation Matrix

From the last row of the column labeled "Proper Euler angles" in Wikipedia's discussion of the rotation matrix, we find,

$Z_1{X}_2{Z}_3$

$=$

$\left[\begin{array}{ccc}(c_1{c}_3-c_2{s}_1{s}_3)& (-c_1{s}_3-c_2{c}_3{s}_1)& (s_1{s}_2)\\ (c_3{s}_1+c_1{c}_2{s}_3)& (c_1{c}_2{c}_3-s_1{s}_3)& (-c_1{s}_2)\\ (s_2{s}_3)& (c_3{s}_2)& (c_2)\end{array}\right]$

The equivalent expression can be found in Professor Berciu's online class notes; it reads,

$\hat{R}(\varphi ,\theta ,\psi )$

$=$

$\left[\begin{array}{ccc}(\cos \varphi \cos \psi -\cos \theta \sin \varphi \sin \psi )& (-\cos \varphi \sin \psi -\cos \theta \cos \psi \sin \varphi )& (\sin \varphi \sin \theta )\\ (\cos \psi \sin \varphi +\cos \varphi \cos \theta \sin \psi )& (\cos \varphi \cos \theta \cos \psi -\sin \varphi \sin \psi )& (-\cos \varphi \sin \theta )\\ (\sin \theta \sin \psi )& (\cos \psi \sin \theta )& (\cos \theta )\end{array}\right]$

See Also

• Wikipedia Chapter on Euler Angles
• Online class notes from Professor Mona Berciu, Department of Physics & Astronomy, University of British Columbia