Euler angles

PD Image
Euler angles. From left to right: initial configuration, after rotation over α, after rotation over β, and after rotation over γ.

In physics, mathematics, and engineering, Euler angles are three rotation angles, commonly denoted by 0 ≤ α ≤ 2π, 0 ≤ β ≤ π, and 0 ≤ γ ≤ 2π. Any rotation of a 3-dimensional object can be performed by three consecutive rotations over the three Euler angles.

Different conventions are in use: a rotation can be active (the object is rotated, the system of axes is fixed in space), or passive (the object is fixed in space, the axes are rotated).

The order of Euler angles may vary; a very common active convention is the z-y′-z′ convention. Attach a system of Cartesian coordinate axes to the body that is to be rotated (the coordinate frame is fixed to the body and is rotated simultaneously with it); in the figure the body-fixed frame is shown in red and labeled by lowercase letters. First rotate around z, then around the new body-fixed y-axis, y′, and finally around z′. Another convention often used is the z-x′-z′ convention, where instead of over the new y-axis the second rotation is over the new x-axis. Also the x-y′-z′′ convention is used.

The right-hand screw rule is practically always followed: the rotation axis is a directed line and a positive rotation is as a cork screw driven into the positive direction of the axis.

Euler angles are used in many different branches of physics and engineering. The present article is written from the point of view of molecular physics, where the objects to be rotated are molecules.

Geometric discussion

In the figure the space-fixed (laboratory) axes are labeled by capital X, Y, and Z and are shown in black. The body to be rotated is not shown, but a system of axes fixed to it is shown in red. One may use any convenient orthonormal frame as a body-fixed frame. Often the body-fixed axes are principal axes, that is, they are eigenvectors of the inertia tensor of the body. Also symmetry axes, when present, may be used. When the body has symmetry axes, the principal axes often coincide with these.

The z-y′-z′ convention will be followed. Initially, the two frames coincide, and the path to a final arbitrary orientation of the body—and its frame—is depicted. The first rotation is around the z-axis, which coincides with the Z-axis. The x- and y-axis move in the circle perpendicular to the z-axis over an angle α. The second rotation is around a circle perpendicular to the y′-axis. The angle is β. It is a great practical advantage of the present convention that the z′-axis has the usual spherical polar coordinates α ≡ φ and β ≡ θ with respect to the space-fixed frame. (Due to this the Wigner D-matrix elements are equal to spherical harmonics for γ = 0).

The final rotation is in a circle perpendicular to the z′-axis over an angle γ. From geometric considerations follows that any orientation in space of the body-fixed frame may be obtained.

Let us write $\mathbf {R} (\varphi ,{\hat {n}})$ for the rotation matrix that describes a rotation around the unit vector ${\hat {n}}$ over an angle $\varphi$. In this article it is shown that if A is a rotation matrix, that

$\mathbf {R} (\varphi ,\mathbf {A} {\hat {n}})=\mathbf {A} \mathbf {R} (\varphi ,{\hat {n}})\mathbf {A} ^{\mathrm {T} }$

where the superscript T indicates the transpose of the matrix. For rotation matrices it holds that the transposed matrix is equal to the inverse of the matrix.

Clearly then the three consecutive rotations correspond to the matrix product

$\mathbf {R} (\gamma ,{\hat {e}}_{z'})\mathbf {R} (\beta ,{\hat {e}}_{y'})\mathbf {R} (\alpha ,{\hat {e}}_{z})$

where unit vectors along the body-fixed axes z, y′, and z′ are introduced. Further

$\mathbf {R} (\beta ,{\hat {e}}_{y'})=\mathbf {R} (\alpha ,{\hat {e}}_{z})\mathbf {R} (\beta ,{\hat {e}}_{y})\mathbf {R} (\alpha ,{\hat {e}}_{z})^{\mathrm {T} }$

and

$\mathbf {R} (\gamma ,{\hat {e}}_{z'})=\mathbf {R} (\beta ,{\hat {e}}_{y'})\mathbf {R} (\gamma ,{\hat {e}}_{z})\mathbf {R} (\beta ,{\hat {e}}_{y'})^{\mathrm {T} }$

(To be continued)

Algebraic treatment

A proper rotation matrix R can be factorized thus

$\mathbf {R} =\mathbf {R} _{z}(\omega _{3})\;\mathbf {R} _{y}(\omega _{2})\;\mathbf {R} _{x}(\omega _{1})$

which is referred to as the Euler z-y-x parametrization, or also as

$\mathbf {R} =\mathbf {R} _{z}(\alpha )\;\mathbf {R} _{y}(\beta )\;\mathbf {R} _{z}(\gamma )\quad$

the Euler z-y-z parametrization.

Here the matrices representing rotations around the z, y, and x axis, respectively, over arbitrary angle φ, are

$\mathbf {R} _{z}(\varphi )\equiv {\begin{pmatrix}\cos \varphi &-\sin \varphi &0\\\sin \varphi &\cos \varphi &0\\0&0&1\\\end{pmatrix}},\quad \mathbf {R} _{y}(\varphi )\equiv {\begin{pmatrix}\cos \varphi &0&\sin \varphi \\0&1&0\\-\sin \varphi &0&\cos \varphi \\\end{pmatrix}},\quad \mathbf {R} _{x}(\varphi )\equiv {\begin{pmatrix}1&0&0\\0&\cos \varphi &-\sin \varphi \\0&\sin \varphi &\cos \varphi \\\end{pmatrix}}.$

Proof

First the Euler z-y-x-parametrization will be proved by describing an algorithm for the factorization of R. Consider to that end the matrix product

$\mathbf {R} _{z}(\omega _{3})\,\mathbf {R} _{y}(\omega _{2})={\begin{pmatrix}\cos \omega _{3}\cos \omega _{2}&-\sin \omega _{3}&\cos \omega _{3}\sin \omega _{2}\\\sin \omega _{3}\cos \omega _{2}&\cos \omega _{3}&\sin \omega _{3}\sin \omega _{2}\\-\sin \omega _{2}&0&\cos \omega _{2}\\\end{pmatrix}}\equiv (\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3}).$

The columns of the matrix product are for ease of reference designated by a₁, a₂, and a₃. Note that the multiplication by R_x(ω₁) on the right does not affect the first column, so that a₁ = r₁ (the first column of the matrix to be factorized). Solve $\omega _{2}\;$ and $\omega _{3}\;$ from the first column of R,

$\mathbf {a} _{1}={\begin{pmatrix}\cos \omega _{3}\;\cos \omega _{2}\\\sin \omega _{3}\;\cos \omega _{2}\\-\sin \omega _{2}\\\end{pmatrix}}={\begin{pmatrix}R_{11}\\R_{21}\\R_{31}\\\end{pmatrix}}\equiv \mathbf {r} _{1}.$

This is possible. First solve $\omega _{2}\;$ for $-\pi /2\leq \omega _{2}\leq \pi /2$ from

$\sin \omega _{2}=-R_{31}.\,$

Then solve $\omega _{3}\;$ for $0\leq \omega _{3}\leq 2\pi$ from the two equations:

${\begin{aligned}\cos \omega _{3}=&{R_{11} \over \cos \omega _{2}}\\\sin \omega _{3}=&{R_{21} \over \cos \omega _{2}}.\end{aligned}}$

Knowledge of $\omega _{2}\;$ and $\omega _{3}\;$ determines the vectors a₂ and a₃.

Since a₁, a₂ and a₃ are the columns of a proper rotation matrix they form an orthonormal right-handed system. The plane spanned by a₂ and a₃ is orthogonal to $\mathbf {a} _{1}\equiv \mathbf {r} _{1}$ and hence the plane contains $\mathbf {r} _{2}$ and $\mathbf {r} _{3}$. Thus the latter two vectors are a linear combination of the first two,

$(\mathbf {r} _{2},\mathbf {r} _{3})=(\mathbf {a} _{2},\mathbf {a} _{3}){\begin{pmatrix}\cos \omega _{1}&-\sin \omega _{1}\\\sin \omega _{1}&\cos \omega _{1}\\\end{pmatrix}}.$

Since $\mathbf {r} _{2},\;\mathbf {a} _{2},\;\mathbf {a} _{3}$ are known unit vectors we can compute

${\begin{aligned}\mathbf {a} _{2}\cdot \mathbf {r} _{2}=&\cos \omega _{1}\\\mathbf {a} _{3}\cdot \mathbf {r} _{2}=&\sin \omega _{1}.\end{aligned}}$

These equations give $\omega _{1}\;$ with $0\leq \omega _{1}\leq 2\pi$.

Augment the 2×2 matrix to the 3×3 matrix $\mathbf {R} _{x}(\omega _{1})$, then

${\begin{aligned}\mathbf {R} \equiv (\mathbf {r} _{1},\mathbf {r} _{2},\mathbf {r} _{3})=(\mathbf {r} _{1},\mathbf {a} _{2},\mathbf {a} _{3})\mathbf {R} _{x}(\omega _{1})=(\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {a} _{3})\mathbf {R} _{x}(\omega _{1})=\mathbf {R} _{z}(\omega _{3})\,\mathbf {R} _{y}(\omega _{2})\,\mathbf {R} _{x}(\omega _{1}).\end{aligned}}$

This concludes the proof of the z-y-x parametrization.

The Euler z-y-z parametrization is obtained by a small modification of the previous proof. Solve $\omega _{2}\;$ and $\omega _{3}\;$ from $\mathbf {r} _{3}=\mathbf {a} _{3}$ (the rightmost multiplication by R_z(ω₁) does not affect r₃) and then consider

$(\mathbf {r} _{1},\;\mathbf {r} _{2})=(\mathbf {a} _{1},\;\mathbf {a} _{2}){\begin{pmatrix}\cos \omega _{1}&-\sin \omega _{1}\\\sin \omega _{1}&\cos \omega _{1}\\\end{pmatrix}}$

or, $\mathbf {a} _{1}\cdot \mathbf {r} _{1}=\cos \omega _{1}\;,\quad \mathbf {a} _{2}\cdot \mathbf {r} _{1}=\sin \omega _{1}.$ The equation for R can be written as

$(\mathbf {r} _{1},\mathbf {r} _{2},\mathbf {r} _{3})=(\mathbf {a} _{1},\mathbf {a} _{2},\mathbf {r} _{3})\,\mathbf {R} _{z}(\omega _{1})=\mathbf {R} _{z}(\omega _{3})\,\mathbf {R} _{y}(\omega _{2})\,\mathbf {R} _{z}(\omega _{1})\;,$

which proves the Euler z-y-z parametrization. It is common in this parametrization to write

$\omega _{3}=\alpha ,\quad \omega _{2}=\beta ,\quad \omega _{1}=\gamma .$

(To be continued)