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.

Basic Relations

In terms of the unit vectors of the $(X,Y,Z)$ Cartesian coordinate system shown in the left panel of Figure 1, we can uniquely specify the (red) vector, $\overrightarrow{A}$, by the expression,

$\overrightarrow{A}={\overrightarrow{e}}_X(A_X)+{\overrightarrow{e}}_Y(A_Y)+{\overrightarrow{e}}_Z(A_Z)$

where the coefficient triplet, $(A_X,A_Y,A_Z)$, give the length (and $\pm$ direction) of each vector component. Alternatively, in terms of the unit vectors of the $(x_1,x_2,x_3)$ Cartesian coordinate system shown in the right panel of Figure 1, the same (red) vector is specified by the expression,

$\overrightarrow{A}={\overrightarrow{e}}_1(A_1)+{\overrightarrow{e}}_2(A_2)+{\overrightarrow{e}}_3(A_3).$

Figure 1

Now, it is clear from the nature of unit vectors and vector dot-products that the value of the coefficient, $A_1$ — which explicitly appears in the second of these two expressions — may be obtained from the dot product, ${\overrightarrow{e}}_1\cdot \overrightarrow{A}$. The same must be true if we insert, for $\overrightarrow{A}$, the first of the two expressions; that is to say,

$A_1={\overrightarrow{e}}_1\cdot \overrightarrow{A}$

$=$

${\overrightarrow{e}}_1\cdot {\overrightarrow{e}}_X(A_X)+{\overrightarrow{e}}_1\cdot {\overrightarrow{e}}_Y(A_Y)+{\overrightarrow{e}}_1\cdot {\overrightarrow{e}}_Z(A_Z).$

Analogously, we can write,

$A_2={\overrightarrow{e}}_2\cdot \overrightarrow{A}$	$=$	${\overrightarrow{e}}_2\cdot {\overrightarrow{e}}_X(A_X)+{\overrightarrow{e}}_2\cdot {\overrightarrow{e}}_Y(A_Y)+{\overrightarrow{e}}_2\cdot {\overrightarrow{e}}_Z(A_Z),$
$A_3={\overrightarrow{e}}_3\cdot \overrightarrow{A}$	$=$	${\overrightarrow{e}}_3\cdot {\overrightarrow{e}}_X(A_X)+{\overrightarrow{e}}_3\cdot {\overrightarrow{e}}_Y(A_Y)+{\overrightarrow{e}}_3\cdot {\overrightarrow{e}}_Z(A_Z).$

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