Euler (1767 and 1776)

Euler angles are a sequence of three angles representing three successive rotations about a set of orthogonal axes. Several more modern sources incorrectly attribute the introduction of "Euler angles" to Euler's 1776 papers on rigid-body dynamics [34, 35]. In fact, those papers never used Euler angles at all; they used the axis-angle representation and the direction-cosine matrix. Euler actually introduced what we would now call Euler angles in his 1767 paper "Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile" ("On the motion of an arbitrary rigid body rotating about a moving axis") [36].

This paper was important because Euler fully distinguished between a space-fixed frame and a body-fixed frame. The space-fixed frame was viewed in reference to the "celestial sphere", with its origin $O$ at the center of gravity of the body. Euler's description was ambiguous, but he did include a figure, reproduced here:

Euler's Figure 2. This reproduces a figure from Euler's 1767 paper [36], showing his conventions for angles describing the orientation of a rigid body in space — which we now know as Euler angles. He also used spherical coordinates for a point relative to the body.

From the figure, we can see that he introduced an orthogonal set of axes $OA$, $OD$, and $OC$ corresponding to what we would now label as $x$, $y$, and $z$ as a right-handed coordinate system fixed in space. He also assigned the axis $OM$ to be the body's principal axis, which we would label as the body-fixed axis $z''$. While he didn't introduce a full body-fixed frame at this point,^[1] he did discuss a reference plane spanned by $MOL$. We can think of this as the rotated plane spanned by $COB$, where $L$ is simply the point at which this rotated plane intersects the "horizon" plane spanned by $AOD$. Thus, $MOL$ tracks the rotation of the body about its axis.

Finally, Euler defines a set of three angles, which we can interpret as corresponding to three successive rotations. Here we translate from his original:

• the angle $ACM$ or the arc $AP$ = $p$,
• the distance of the point $M$ from the zenith $C$ or the arc $CM$ = $q$,
• and the angle $CML$ = $r$.

We readily see that these correspond roughly to our angles $α$, $β$, $γ$, respectively — with the important exception that the origin of $γ$ is different from the origin of $r$. Specifically, since $r$ is measured by $CML$, $r=0$ when the reference direction points back toward the zenith $C$ from $M$. This is opposite to modern conventions, where the reference direction points toward the nadir when $γ=0$. So, putting it all together, we have:

\[\begin{align*} α &= p, \\ β &= q, \\ γ &= π - r. \end{align*}\]

Euler then goes on to define the position of an arbitrary point, labeled $Z$ in the figure, by what we would now call Cartesian coordinates relative to the fixed frame, and also spherical coordinates relative to the body frame. First, he progresses from the origin $O$ along the $OA$ axis to the point $X$, then parallel to $OD$ to the point $Y$, and finally parallel to $OC$ to the point $Z$. He specifically lists the distances $OX=x$, $XY=y$, and $YZ=z$, which are precisely what we would label them as. Then, he defines the distance $OZ$ as $s$, the arc $MN$ as $u$, and the angle $LMN$ as $v$. That is, we would denote these quantities as

\[\begin{align*} r &= s, \\ θ &= u, \\ ϕ &= v, \end{align*}\]

but only in the body-fixed frame.

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