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, $\vec{A}$ , by the expression,

$\vec{A} = {\vec{e}}_{X} (A_{X}) + {\vec{e}}_{Y} (A_{Y}) + {\vec{e}}_{Z} (A_{Z})$

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,

$\vec{A} = {\vec{e}}_{1} (A_{1}) + {\vec{e}}_{2} (A_{2}) + {\vec{e}}_{3} (A_{3}) .$

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, ${\vec{e}}_{1} \cdot \vec{A}$ . The same must be true if we insert, for $\vec{A}$ , the first of the two expressions; that is to say,

$A_{1} = {\vec{e}}_{1} \cdot \vec{A}$

=

${\vec{e}}_{1} \cdot {\vec{e}}_{X} (A_{X}) + {\vec{e}}_{1} \cdot {\vec{e}}_{Y} (A_{Y}) + {\vec{e}}_{1} \cdot {\vec{e}}_{Z} (A_{Z}) .$

Analogously, we can write,

$A_{2} = {\vec{e}}_{2} \cdot \vec{A}$	$=$	${\vec{e}}_{2} \cdot {\vec{e}}_{X} (A_{X}) + {\vec{e}}_{2} \cdot {\vec{e}}_{Y} (A_{Y}) + {\vec{e}}_{2} \cdot {\vec{e}}_{Z} (A_{Z}),$
$A_{3} = {\vec{e}}_{3} \cdot \vec{A}$	$=$	${\vec{e}}_{3} \cdot {\vec{e}}_{X} (A_{X}) + {\vec{e}}_{3} \cdot {\vec{e}}_{Y} (A_{Y}) + {\vec{e}}_{3} \cdot {\vec{e}}_{Z} (A_{Z}) .$

This set of three relations therefore provides a generic way to express one set of projections in terms of the other. When written in matrix form, the set becomes,

$[\begin{matrix} A_{1} \\ A_{2} \\ A_{3} \end{matrix}]$

$=$

$[\begin{matrix} {\vec{e}}_{1} \cdot {\vec{e}}_{X} & {\vec{e}}_{1} \cdot {\vec{e}}_{Y} & {\vec{e}}_{1} \cdot {\vec{e}}_{Z} \\ {\vec{e}}_{2} \cdot {\vec{e}}_{X} & {\vec{e}}_{2} \cdot {\vec{e}}_{Y} & {\vec{e}}_{2} \cdot {\vec{e}}_{Z} \\ {\vec{e}}_{3} \cdot {\vec{e}}_{X} & {\vec{e}}_{3} \cdot {\vec{e}}_{Y} & {\vec{e}}_{3} \cdot {\vec{e}}_{Z} \end{matrix}] \cdot [\begin{matrix} A_{X} \\ A_{Y} \\ A_{Z} \end{matrix}] = \hat{R} \cdot [\begin{matrix} A_{X} \\ A_{Y} \\ A_{Z} \end{matrix}] .$

Alternatively, we could "dot" the unit-vector triplet $({\vec{e}}_{X}, {\vec{e}}_{Y}, {\vec{e}}_{Z})$ into the vector, $\vec{A}$ , in which case it would be easy to demonstrate that mapping the other direction is accomplished via the relation,

$[\begin{matrix} A_{X} \\ A_{Y} \\ A_{Z} \end{matrix}]$

$=$

${\hat{R}}^{- 1} \cdot [\begin{matrix} A_{1} \\ A_{2} \\ A_{3} \end{matrix}],$

where,

${\hat{R}}^{- 1}$

$\equiv$

$[\begin{matrix} {\vec{e}}_{X} \cdot {\vec{e}}_{1} & {\vec{e}}_{X} \cdot {\vec{e}}_{2} & {\vec{e}}_{X} \cdot {\vec{e}}_{3} \\ {\vec{e}}_{Y} \cdot {\vec{e}}_{1} & {\vec{e}}_{Y} \cdot {\vec{e}}_{2} & {\vec{e}}_{Y} \cdot {\vec{e}}_{3} \\ {\vec{e}}_{Z} \cdot {\vec{e}}_{1} & {\vec{e}}_{Z} \cdot {\vec{e}}_{2} & {\vec{e}}_{Z} \cdot {\vec{e}}_{3} \end{matrix}] .$

A Sequence of Rotations

It is quite generally true that we can transition/map/migrate from one set of orthogonal unit vectors — such as the $({\vec{e}}_{X}, {\vec{e}}_{Y}, {\vec{e}}_{Z})$ inertial/laboratory frame illustrated by the black arrows in the left panel of Figure 1 — to any other set of orthogonal unit vectors — such as the $({\vec{e}}_{1}, {\vec{e}}_{2}, {\vec{e}}_{3})$ body frame illustrated by the black arrows in the right panel of Figure 1 — by carrying out three rotations. A fairly standard sequence of rotations is depicted in the three panels of Figure 2:

Rotate the triplet of unit vectors about the $Z$ (i.e., ${\vec{e}}_{Z}$ ) axis by an angle, $ϕ$ . The result is the green coordinate system labeled, $(1^{'}, 2^{'}, 3^{'})$ . Note: (a) The 3' axis remains aligned with the inertial-frame Z-axis; and (b) we will refer to the 1' axis as the line of nodes.
Rotate the triplet of unit vectors about the line of nodes by an angle, $θ$ . The result is the light-blue coordinate system labeled, (1", 2", 3"). Note that the 1" axis is aligned with the 1' axis (line of nodes).

Figure 2

Rotation #1	Rotation #2	Rotation #3

Rotation Matrix

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

$Z_{1} X_{2} Z_{3}$

$=$

$[\begin{matrix} (c_{1} c_{3} - c_{2} s_{1} s_{3}) & (- c_{1} s_{3} - c_{2} c_{3} s_{1}) & (s_{1} s_{2}) \\ (c_{3} s_{1} + c_{1} c_{2} s_{3}) & (c_{1} c_{2} c_{3} - s_{1} s_{3}) & (- c_{1} s_{2}) \\ (s_{2} s_{3}) & (c_{3} s_{2}) & (c_{2}) \end{matrix}]$

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

$\hat{R} (ϕ, θ, ψ)$

$=$

$[\begin{matrix} (\cos ϕ \cos ψ - \cos θ \sin ϕ \sin ψ) & (- \cos ϕ \sin ψ - \cos θ \cos ψ \sin ϕ) & (\sin ϕ \sin θ) \\ (\cos ψ \sin ϕ + \cos ϕ \cos θ \sin ψ) & (\cos ϕ \cos θ \cos ψ - \sin ϕ \sin ψ) & (- \cos ϕ \sin θ) \\ (\sin θ \sin ψ) & (\cos ψ \sin θ) & (\cos θ) \end{matrix}]$

Appendix/Mathematics/EulerAngles

Contents

Euler Angles

Basic Relations

A Sequence of Rotations

Rotation Matrix

See Also

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Appendix/Mathematics/EulerAngles

Euler Angles

Basic Relations

A Sequence of Rotations

Rotation Matrix

See Also

Navigation menu

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