Geodesics of the rotation group SO(3)

A geodesic is a curve of shortest distance between two points on a manifold (surface). Classic examples include the geodesic between two points in a Euclidean space is a straight line and the geodesic between two points on a sphere is a great circle. Dating to Jacobi in the 1800s, it is known that for two points on a manifold $\mathcal{S}$ , a necessary condition for a geodesic between two points can be found by imagining a particle moving on $\mathcal{S}$ (see, e.g., Lanczos [1]). The necessary condition is identical to the motion of the particle satisfying Lagrange’s equations of motion for the particle. That is, the path $\mathcal{C}$ of the particle is a geodesic for pairs of points that $\mathcal{C}$ passes through.

With respect to the simplest possible metric (measure of distance) for the group of rotations $SO(3)$ , geodesics of $SO(3)$ are characterized by constant angular velocity motions of rigid bodies and, if quaternions are used to parameterize the rotation, as great circles on a three-sphere (cf. Figure 1). As discussed below, the former interpretation is employed used in optometry to characterize saccadic motions of the eye (see [2, 3, 4, 5, 6, 7, 8, 9, 10]) and the latter is the basis for the SLERP algorithm interpolation of rotations in computer graphics (see Shoemake [11]). The simplicity of these two disparate interpretations belies the complexity of the corresponding rotations. By using a quaternion representation for a rotation, a simple proof of the equivalence of the aforementioned characterizations and a straightforward method to establish features of the corresponding rotations are discussed. Our discussion is based on the recent papers by Novelia and O’Reilly [12, 13] and O’Reilly and Payen [14].

**Figure 1.** Three distinct representations of a class of constant angular velocity motion. From left to right, a representation using the components of ${\bf R}$ and Steiner’s Roman Surface, a representation using a unit quaternion on a two-sphere, and a representation using the corotational basis vectors $\left\{{\bf e}_1, {\bf e}_2, {\bf e}_3\right\}$ of a rigid body motion. This figure is adopted from Novelia and O’Reilly [13].

The choice of the simplest metric is equivalent to examining the moment-free rotational motions of a spherically symmetric rigid body. For such a rigid body, the moment of inertia tensor ${\bf J}$ is simply a scalar multiple of the identity and, using a balance of angular momentum, ${\bf M} = d/dt\left( {\bf J}\bomega\right)$ , we can conclude that the angular velocity vector $\bomega$ is constant for all moment-free motions. Thus, if you throw a tennis ball into the air, and ignore drag, then it will rotate with a constant angular velocity. The rotation tensor ${\bf Q}(t)$ of the ball describes a geodesic on $SO(3)$ with respect to the simplest metric. There are several methods to visualize the rotation: components of a quaternion on a three-sphere and an immersion using Steiner’s Roman surface. The immersion was first described by Apery [15] and can (surprisingly) be related to the components of the rotation tensors ${\bf Q}$ .

By changing the metric, the geodesics change. To observe this phenomenon, we again exploit the analogy with moment-free motion of a rigid body and consider axisymmetric and asymmetric rigid bodies. We again find that a subset of the geodesics correspond to motions with constant angular velocities, but also observe that more complex rotational motions of the rigid body can also serve as geodesics of $SO(3)$ .

Contents

The Euler-Rodrigues and quaternion parameterizations

The use of four Euler-Rodrigues symmetric (or Euler symmetric) parameters to parameterize a rotation dates to Euler [16] in 1771 and Rodrigues [17] in 1840 [18, 19, 20]. We denote these parameters by the pair $\left(\beta_0, \, \bbeta\right)$ , where $\beta_0$ is a scalar and $\bbeta$ is a vector. In 1843, Hamilton [21] made his discovery of quaternion multiplication, and shortly afterwards Cayley [22] published results showing how quaternions could be used to parameterize a rotation. The historical development of these parameterizations features some of the greatest mathematicians of the 18th and 19th centuries. These individuals include Euler, Gauss [23], and Cayley, among others. The topic also provides a fascinating introduction to this period. For further details, we refer the reader to the works [18, 19, 20, 24, 25]. The use of quaternions to parameterize saccadic motions of the eye was successfully championed by Westheimer [10].

The Euler-Rodrigues symmetric parameters $\beta_0$ and $\bbeta$ satisfy the constraint

(1) $\begin{equation*}\beta^2_0 + \bbeta \cdot \bbeta = 1.\end{equation*}$

Let the pair $\left(q_0, \, {\bf q}\right)$ , where $q_0$ is a scalar and ${\bf q}$ is a vector, denote a unit quaternion, which has unit norm:

(2) $\begin{equation*}q^2_0 + {\bf q} \cdot {\bf q} = 1.\end{equation*}$

Thus, a unit quaternion can be used to define a set of Euler-Rodrigues symmetric parameters, and vice versa: $\beta_0 = q_0$ and $\bbeta = {\bf q}$ . In the four-dimensional space parameterized by the components of a quaternion, the set of all unit quaternions defines a unit sphere referred to as the 3-sphere $S^3$ . The parameters $\beta_0 = q_0$ and $\bbeta = {\bf q}$ can be used to define a rotation about an axis ${\bf r}$ through an angle $\phi$ using the identifications

(3) $\begin{eqnarray*}&& \beta_0 = q_0 = \cos\left( \frac{\phi}{2} \right),\\\\\\&& \bbeta = {\bf q} = \sin\left( \frac{\phi}{2} \right){\bf r}.\end{eqnarray*}$

The resulting representation of the rotation tensor ${\bf R}$ , which transforms the fixed basis $\left\{ {\bf E}_1, \, {\bf E}_2, \, {\bf E}_3\right\}$ into the corotational basis $\left\{ {\bf e}_1, \, {\bf e}_2, \, {\bf e}_3\right\}$ , is given by

(4) $\begin{equation*}{\bf R} = {\bf R}\left(q_0, \, {\bf q}\right) = \left(q^2_0 - {\bf q}\cdot{\bf q}\right){\bf I} + 2{\bf q}\otimes{\bf q} - 2q_0\left(\bepsilon{\bf q}\right).\end{equation*}$

Examining the tensor components $R_{ik} = {\bf e}_k\cdot{\bf E}_i$ , one finds that the components are quadratic functions of the four parameters $q_0$ and ${\bf q}$ :

(5) $\begin{equation*}\mathsf{R} =\left[ \begin{array}{c c c}R_{11} & R_{12} & R_{13} \\R_{21} & R_{22} & R_{23} \\R_{31} & R_{32} & R_{33}\end{array} \right] = \left( 2 q^2_0 - 1 \right)\left[ \begin{array}{c c c}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array} \right]+ 2\left[ \begin{array}{c c c}q^2_1 & q_1q_2 & q_1q_3 \\q_1q_2 & q^2_2 & q_2q_3 \\q_1q_3 & q_2q_3 & q^2_3\end{array} \right]+ 2q_0\left[ \begin{array}{ccc}0 & -q_3 & q_2 \\q_3 & 0 & -q_1 \\-q_2 & q_1 & 0\end{array}\right] . \hspace{1in} \scalebox{0.001}{\textrm{\textcolor{white}{.}}}\end{equation*}$

Consequently, representation (4) is free of trigonometric functions and singularities, making it attractive from a numerical implementation standpoint. In addition, the fact that unit quaternions lie on the 3-sphere and can be used to represent rotations is exploited in computer graphics to interpolate animations (see Shoemake [11]). Examining (5) in closer detail,

(6) $\begin{equation*}R_{ik} =\left[ \begin{array}{cccc}q_0 & q_1 & q_2 & q_3\end{array} \right]\mathsf{F}_{ik}\left[ \begin{array}{c}q_0 \\q_1 \\q_2 \\q_3\end{array}\right] ,\end{equation*}$

where each of the nine 4 $\times$ 4 matrices $\mathsf{F}_{ik}$ are symmetric and proper-orthogonal. For example,

(7) $\begin{eqnarray*}&& \mathsf{F}_{11} =\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1\end{array}\right],\\\\\\&& \mathsf{F}_{31} =\left[\begin{array}{cccc}0 & 0 & -1 & 0 \\0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 \\0 & 1 & 0 & 0\end{array}\right].\end{eqnarray*}$

The matrices $\mathsf{F}_{ik}$ play a role in constructing stiffness matrices [26] and in establishing identities for the derivatives of the components of the matrix $\mathsf{R}$ [27].

In terms of $q_0$ and ${\bf q}$ , the angular velocity vector $\bomega_{\bf R}$ associated with ${\bf R}$ has the well-known representation

(8) $\begin{equation*}{\bomega}_{\bf R} = 2 \left(q_0\dot{\bf q} - \dot{q}_0{\bf q} + {\bf q}\times\dot{\bf q}\right).\end{equation*}$

It is interesting to note that the time derivative of ${\bf q}$ and its corotational rate $\corot{\bf q}$ are simply related [26]:

(9) $\begin{equation*}\dot{\bf q} = {\bf R}^T \corot{\bf q}.\end{equation*}$

To verify this result, recall that ${\bf q} = \sum_{k \, \, = \, 1}^3 q_i {\bf E}_i = \sum_{i \, \, = \, 1}^3 q_i {\bf e}_i$ because ${\bf R}{\bf q} = {\bf q}$ . Computing $\corot{\bf q} = \sum_{i \, \, = \, 1}^3 \dot{q}_i {\bf e}_i$ and then using the identity ${\bf R}^T{\bf e}_i = {\bf E}_i$ confirms (9). With the help of (9), some lengthy but straightforward manipulations of (8) reveal that $\bomega_{\bf R}$ has the equivalent representation

(10) $\begin{equation*}\bomega_{\bf R} = 2 \left ( q_0 \corot{\bf q} - \dot{q}_0 {\bf q} - {\bf q} \times \corot{\bf q} \right ) \end{equation*}$

For completeness, we note that the angular velocity $\bomega_{0_{\bf R}}$ , defined as the axial vector of ${\bf R}^T\dot{\bf R}$ , can be obtained from $\bomega_{\bf R}$ according to

(11) $\begin{equation*}\bomega_{0_{\bf R}} = {\bf R}^T \bomega_{\bf R}.\end{equation*}$

Using (10) and the identities ${\bf R}{\bf q} = {\bf q}$ , (9), and ${\bf R}^T\left({\bf a}\times{\bf b}\right) = \left({\bf R}^T{\bf a}\times{\bf R}^T{\bf b}\right)$ for any vectors ${\bf a}$ and ${\bf b}$ , it follows that $\bomega_{0_{\bf R}}$ has the representation

(12) $\begin{equation*}{\bomega}_{0_{\bf R}} = 2 \left ( q_0 \dot{\bf q} - \dot{q}_0 {\bf q} - {\bf q} \times \dot{\bf q} \right ) \end{equation*}$

As discussed in [28, 29], this representation is convenient to use when computing Lagrange’s equations of motion for rigid bodies. Lastly, taking the ${\bf e}_i$ components of representation (10) for $\bomega_{\bf R}$ leads to

(13) $\begin{equation*}\left[ \begin{array}{c}\bomega_{\bf R}\cdot{\bf e}_1 \\\bomega_{\bf R}\cdot{\bf e}_2 \\\bomega_{\bf R}\cdot{\bf e}_3\end{array} \right] = \mathsf{C}\left[ \begin{array}{c}\dot{q}_0 \\\dot{q}_1 \\\dot{q}_2 \\\dot{q}_3\end{array} \right],\end{equation*}$

where

(14) $\begin{equation*}\mathsf{C} = 2\left[ \begin{array}{c c c c}-q_1 & q_0 & q_3 & -q_2 \\-q_2 & -q_3 & q_0 & q_1 \\-q_3 & q_2 & -q_1 & q_0\end{array} \right].\end{equation*}$

The $3 \times 4$ matrix $\mathsf{C}$ has the generalized inverse

(15) $\begin{equation*}\mathsf{C}^{-} = \frac{1}{4}\mathsf{C}^T .\end{equation*}$

That is, $\mathsf{C}^{-} \mathsf{C} = \mathsf{I}$ , where $\mathsf{I}$ is the 4 $\times$ 4 identity matrix. Thus, using $\mathsf{C}^{-}$ , (13) can be rearranged to establish the following differential equations for the parameters $q_0$ and ${\bf q}$ in terms of $\omega_i = \bomega_{\bf R}\cdot{\bf e}_i$ :

(16) $\begin{equation*}\left[ \begin{array}{c}\dot{q}_0 \\\dot{q}_1 \\\dot{q}_2 \\\dot{q}_3\end{array} \right] = \frac{1}{2}\left[ \begin{array}{c c c}- q_1 & -q_2 & -q_3 \\q_0 & - q_3 & q_2 \\q_3 & q_0 & - q_1 \\-q_2 & q_1 & q_0\end{array} \right]\left[ \begin{array}{c}\omega_1 \\\omega_2 \\\omega_3\end{array} \right].\end{equation*}$

Given measurements for $\omega_i(t)$ and the initial orientation $q_0\left(t_0\right)$ and ${\bf q}\left(t_0\right)$ of a body, these differential equations can be integrated to determine $q_0\left(t\right)$ and ${\bf q}\left(t\right)$ , and hence the body’s orientation ${\bf R}(t)$ .

Geodesics of SO(3): the simplest metric and the dynamics of a spherically symmetric body

The shortest distance between two points on a manifold depends on the measure of distance used on the manifold. The simplest metric for $SO(3)$ is based on the inner-product $\bomega\cdot\bomega$ where $\bomega$ is the angular velocity vector associated with the rotation tensor ${\bf Q} \in SO(3)$ . This choice of metric is equivalent to examining the moment-free motion of a spherically symmetric rigid body.

For a spherically symmetric rigid body of mass $m$ and radius $R = \sqrt{\frac{5}{2}}$ , the rotational kinetic energy of the rigid body is simply $T = \frac{m}{2} \bomega\cdot\bomega$ where $\bomega$ is the angular velocity vector of the rigid body and ${\bf Q}$ is the associated rotation tensor. We can use this energy to define the kinematical line-element (or measure of distance) $ds$ for $SO(3)$ :

(17) $\begin{equation*}ds = \sqrt{\frac{2T}{m}} dt = \sqrt{\bomega\cdot\bomega} dt.\end{equation*}$

Here,

(18) $\begin{eqnarray*}\bomega\cdot\bomega &=& 4 \dot{q}^2_0 + 4 \dot{\bf q}\cdot\dot{\bf q}\nonumber \\&=& 4 \dot{q}^2_0 + 4 \dot{q}^2_1 + 4 \dot{q}^2_2 + 4\dot{q}^2_3.\end{eqnarray*}$

The geodesics with respect to $ds$ are extremizers of $T$ and, appealing to Jacobi’s theorem, we note that the rotational kinetic energy $T$ is conserved along the geodesics (see, e.g., Lanczos [1]). Conservation of $T$ also implies that the angular speed $\sqrt{\bomega\cdot\bomega}$ of the rotational motion that corresponds to the geodesic is constant.

If we use unit quaternions to parameterize ${\bf Q}$ , then a two-to-one covering of $SO(3)$ is obtained. That is, the pair of quaternions $\left(q_0, {\bf q}\right)$ and $\left(- q_0, - {\bf q}\right)$ , correspond to the same rotation. Unit quaternions can be considered as points on a 3-dimensional sphere $S^3$ that is embedded in a 4-dimensional Euclidean space. To explore the geodesics on $SO(3)$ , it is first convenient for illustrative purposes to consider geodesics of $S^3$ using the kinematical line-element $ds$ . The necessary conditions for $\left(q_0(t), {\bf q}(t)\right)$ to correspond to a geodesic are established using a variational principle for geodesics on a configuration manifold. The resulting conditions are the Euler-Lagrange necessary condition:

(19) $\begin{equation*}\frac{d}{dt}\left( \frac{\partial T}{\partial \dot{q}_K} \right) - \frac{\partial T}{\partial {q}_K} = \lambda_1 q_K, \qquad\left(K = 0, 1,2,3\right),\end{equation*}$

where $\lambda_1$ is a Lagrange multiplier associated with the Euler parameter constraint:

(20) $\begin{equation*}1 = q^2_0 + q^2_1 + q^2_2 + q^2_3.\end{equation*}$

Substituting for $T$ , (19) reduce to

(21) $\begin{equation*}4 m \ddot{q}_K = C q_K, \qquad\end{equation*}$

where $C=\lambda_1$ . These equations have the integral of motion $T$ and we denote the value of this integral of motion by $T_0$ . While our construction of (21) follows from a variational principle for geodesics on a configuration manifold, the corresponding equations of motion also follow from those for the rotational motion of a rigid body whose rotation is parameterized by Euler parameters.

With the help of (20), we follow [27, 29] and solve for $\lambda_1$ :

(22) $\begin{equation*}\lambda_1 = - 2 T_0.\end{equation*}$

Hence, (19) reduce further to

(23) $\begin{equation*}\ddot{q}_K + \left(\frac{T_0}{2m}\right) q_K = 0, \qquad\left(K = 0, 1,2,3\right).\end{equation*}$

These equations imply that the quaternion components associated with a geodesic execute simple harmonic motions:

(24) $\begin{equation*} q_{K}(t) = q_{K}(0) \cos\left( \nu_{0} t \right) + \frac{{\dot{q}}_{K}(0)}{\nu_{0}} \sin\left( \nu_{0} t \right) \end{equation*}$

where the frequency $\nu_0$ is half the magnitude of $\bomega$ :

(25) $\begin{equation*}\nu^2_0 = \frac{T_0}{2m} = \frac{\bomega\cdot\bomega}{4} .\end{equation*}$

The factor of one half in the angular speed of $q_K(t)$ compared to ${\bf e}_i(t)$ can be attributed to the fact that the unit quaternions provide a two-to-one cover for $SO(3)$ . The initial conditions $q_K(0)$ and $\dot{q}_K(0)$ must satisfy the Euler parameter constraint $q_0(0)q_0(0) + {\bf q}(0)\cdot{\bf q}(0) = 1$ and its differential counterpart $q_0(0)\dot{q}_0(0) + {\bf q}(0)\cdot\dot{\bf q}(0) = 0$ .

While the angular speed $\sqrt{\bomega\cdot\bomega}$ associated with the geodesic motions is constant, this does not guarantee that $\bomega$ is a constant. However, as

(26) $\begin{equation*}\dot{\bomega} = 2 q_0 \ddot{\bf q} - 2\ddot{q}_0 {\bf q} + 2{\bf q}\times\ddot{\bf q},\end{equation*}$

and, from (21), $m\ddot{q}_0 = \frac{\lambda_1}{2} q_0$ and $m\ddot{\bf q} = \frac{\lambda_1}{2} {\bf q}$ , we can conclude that $\bomega$ is constant. Thus, geodesics on $S^3$ with respect to $ds$ correspond to rotations with constant angular velocity vectors.

As discussed in [13], the path $q_K(t)$ traced out by the unit quaternion given by (24) are arcs of great circles on the unit 3-sphere. That is, these are the motions used to interpolate rotation tensors in Shoemake’s SLERP algorithm [11].

The simplicity of the differential equations (23) governing $q_K(t)$ , which were first presented in Novelia and O’Reilly [13], enables on to quickly conclude that ${\bf q}$ and $\dot{\bf q}$ lie on a plane by noting the constancy of ${\bf q}\times\dot{\bf q}$ :

(27) $\begin{equation*}\frac{d}{dt}\left({\bf q}\times\dot{\bf q}\right) = {\bf q}\times\ddot{\bf q} = {\bf 0}.\end{equation*}$

An alternative formulation of (23) is presented in O’Reilly and Payen [14]. Their resulting differential equations for the axis ${\bf r}$ and angle $\theta$ of a rotation ${\bf Q}$ corresponding to a constant angular velocity motion are integrable, equivalent to (23) but far more difficult to solve analytically:

(28) $\begin{gather*} \ddot{\theta} = b^2 \sin\left(\theta\right), \nonumber \\ \dot{b} = b \dot{\theta}\left( \frac{ - \sin\left(\theta\right)}{1 - \cos\left(\theta\right)}\right). \end{gather*}$

We invite the reader to compare (28) to the corresponding set of equations expressed using quaternions (23).

In the sequel, and without loss in generality, we often take advantage of the result that ${\bf q}$ and $\dot{\bf q}$ lie on a plane by choosing $\left\{ {\bf E}_1, {\bf E}_2, {\bf E}_3\right\}$ such that ${\bf E}_1$ is normal to the aforementioned plane: $q_1(0) = \dot{q}_1(0) = 0$ . In this case, we can define the angle $\psi$ such that

(29) $\begin{equation*}q_0 = \cos\left(\frac{\theta}{2}\right), \qquad q_2 = \sin\left(\frac{\theta}{2}\right)\cos(\psi), \qquad q_3 = \sin\left(\frac{\theta}{2}\right)\sin(\psi).\end{equation*}$

With the help of (24) it is straightforward to write down analytical expressions for $\theta(t)$ and ${\bf r}(t)$ :

(30) $\begin{equation*}\theta(t) = 2 \cos^{-1}\left(q_0(t)\right), \quad\psi(t) = \tan^{-1}\left(\frac{q_2(t)}{q_3(t)}\right), \quad{\bf r}(t) = \cos(\psi(t)){\bf E}_1 + \sin(\psi(t)){\bf E}_2.\end{equation*}$

The reader should observe that ${\bf r}$ and $\dot{\theta}$ are not necessarily constant even though $\bomega$ is. With reference to the ordinary differential equations (28), $b = \dot{\psi}$ .

Geodesics of SO(3): the simplest metric and the immersion of the real projective plane

Allowing for the fact that we can choose $q_1 = 0$ without loss in generality, the remaining coordinates to parameterize $SO(3)$ are $q_2$ , $q_3$ and $q_0^2 = 1-q_2^2-q_3^2$ . We know that the geodesics of $SO(3)$ are circles on a two-sphere $S^2$ in $\euclid^3$ . The sphere, however, still retains the two-to-one covering property between a pair of antipodal points and a rotation ${\mathbf Q}$ : $\left(q_0, q_2, q_3\right)$ and its antipodal point $\left(- q_0, - q_2, - q_3\right)$ describe the same rotation ${\mathbf Q}$ . It is a well-known result in topology that the quotient space of $S^2$ obtained by identifying antipodal points is the real projective plane $\mathbb{RP}^2$ . Consequently, points in $\mathbb{RP}^2$ are isomorphic to $SO(3)$ when $q_1 = 0$ (cf. [13, 15, 28, 30]). Unfortunately, $\mathbb{RP}^2$ can be embedded in $\euclid^4$ but not $\euclid^3$ . See, for example, Section 7 of the textbook by Tu [31] for a good discussion on projective spaces. O’Reilly’s exposition on these matters in Section 6 of [28] is heavily influenced by the discussion of $\mathbb{RP}^3$ in Tu [31].

To proceed further, we immerse $\mathbb{RP}^2$ into a non-orientable surface in $\euclid^3$ which is known as Steiner’s Roman surface $\mathcal{M}$ . The transformation required to perform the immersion is discussed in Apery’s seminal work [15]:

(31) $\begin{eqnarray*}F: \left(q_0,\,q_3,\,q_2\right) &\mapsto& \left(q_2 q_3,\, q_2 q_0 ,\, q_3 q_0 ,\, q_2^2 + q_3^2 + q_0^2\right).\end{eqnarray*}$

However, from the representation (5) for the components of a rotation tensor ${\bf R}$ , we find that

(32) $\begin{eqnarray*}2q_2q_3 &=& Q_{32} = R_{32} = {\bf e}_2\cdot{\bf E}_3, \nonumber \\2q_2q_0 &=& Q_{13} = R_{13} = {\bf e}_3\cdot{\bf E}_1, \nonumber \\2q_3q_0 &=& Q_{21} = R_{21} = {\bf e}_1\cdot{\bf E}_2.\end{eqnarray*}$

Thus by projecting into $\euclid^3$ , we arrive at the desired mapping

(33) $\begin{equation*}G: (q_0,\,q_3,\,q_2) \mapsto \left(R_{32},\, R_{13} ,\, R_{21}\right).\end{equation*}$

We next recast the parametric equations for the two-dimensional manifold $\mathcal{M}$ , known as Steiner’s Roman surface, in $\euclid^3$ by representing the quaternion components as a function of azimuthal and zenith angles on $S^2$ : $\{u,v\} = \left\{\psi,\frac{\theta}{2}\right\}$ (cf. (29)). The resulting expressions for the Cartesian coordinates of a point on $\mathcal{M}$ are

(34) $\begin{eqnarray*}x_1(u,v) &=& R_{32} = 2q_2 q_3 = \sin(2u) \sin^2(v) = \sin(2 \psi)\sin^2\left(\frac{\theta}{2}\right),\\x_2(u,v) &=& R_{13} = 2q_2q_0 = \cos(u) \sin(2v) = \cos(\psi)\sin\left(\theta\right),\\x_3(u,v) &=& R_{21} = 2q_0q_3 = \sin(u) \sin(2v) = \sin(\psi)\sin\left(\theta\right).\end{eqnarray*}$

Although $\mathcal{M}$ is a non-orientable surface in $\euclid^3$ , all but three rotations can be identified with a single point on $\mathcal{M}$ . The exceptions are the three rotations $\left({\bf r} = {\bf E}_2, \theta = \pi\right)$ , $\left({\bf r} = {\bf E}_3, \theta = \pi\right)$ , and the identity ${\bf Q} = {\bf I}$ . Each of these three rotations are mapped to the origin (also known as the triple point or pinch point of $\mathcal{M}$ ). Because the rotations associated with the triple point are not unique, it is possible to have a closed trajectory starting from and ending at the triple point where the body does not return to its starting orientation.

Representative examples of geodesics of $SO(3)$ as they manifest as curves on $\mathcal{M}$ are shown in Figures 2, 3, and 4. As discussed below, the representations of the geodesics of $SO(3)$ on $\mathcal{M}$ are either straight lines or ellipses.

Representations of the geodesics of SO(3): the simplest metric and the rotation of a rigid body

The geodesic on $SO(3)$ when the simplest metric is employed correspond to rotational motions of a rigid body with a constant angular velocity vector:

(35) $\begin{equation*}\bomega = \bomega_0 = \omega_{1_0}{\bf e}_1 + \omega_{2_0}{\bf e}_2 + \omega_{3_0}{\bf e}_3,\end{equation*}$

where $\omega_{i_0}$ are constant. Consequently, the instantaneous axis ${\bf i}$ is constant. However, these results in and of themselves do not provide the full picture of the behavior of the rotation ${\bf Q}$ . We now examine different manifestations of rotations with constant angular velocities. An important point to note here is that the rotation tensors ${\bf Q}$ and ${\bf Q}{\bf Q}_0$ where ${\bf Q}_0$ is constant have the same angular velocity vectors but may have distinct angles and axes of rotation. This is equivalent to the fact that two motions of a particle may have the same velocity vector but they can have distinct position vectors.

Substituting the expressions (24) for $q_K(t)$ into (5) , the resulting rotation tensor and ${\bf e}_i(t)$ can be found. It is easier however to present some qualitative observations. First, we note that $\bomega$ is constant for the geodesics and hence the projection of ${\bf e}_i$ onto $\bomega$ is constant. Because the magnitude of ${\bf e}_i$ is 1, we can conclude that ${\bf e}_i(t)$ describes an arc of a circle during its motion. The circle in question is centered at a point along the instantaneous axis of rotation ${\bf i} = \bomega/\lnorm \bomega \rnorm$ .

Because of the definitions of quaternion components $q_0, \bold{q}$ , we immediately conclude from (24) that the period of $\theta(t)$ is half that of the quaternion components $q_K(t)$ while the period of ${\bf r}$ is the same as those for the quaternion components. Thus, $\dot{\theta} = \pm \lnorm \bomega_0 \rnorm$ . To gain further insight, we appeal to the identity relating $\dot{\bf r}$ to ${\bf q}$ :

(36) $\begin{equation*}\dot{\bf r} = \frac{1}{\lnorm {\bf q}\rnorm}\left(\dot{\bf q} + \frac{q_0 \dot{q}_0}{\lnorm {\bf q}\rnorm^2} {\bf q} \right).\end{equation*}$

This expression, along with (23) (i.e., $\ddot{q}_K = - \left(\frac{T_0}{2m}\right) q_K$ ) can also be used to establish an expression for $\ddot{\bf r}$ . It is easy to then show that $\left({\bf r}\times\dot{\bf r}\right)\cdot\ddot{\bf r} = 0$ . This result, which was first established in O’Reilly and Payen [14], implies that either ${\bf r}$ is constant or it traces a great circle.

Choosing ${\bf E}_1$ to be normal to ${\bf r}$ (without loss of generality), we previously arrived at the analytical expressions (30) for ${\bf r}(t)$ and $\theta(t)$ . Consequently, we appeal to [14] who found that rotations with constant angular velocity motions can be classified into three types:

Type I Motions where the axis of rotation is parallel to $\bomega$ and $\dot{\theta}$ is a non-zero constant: ${\bf q}\parallel{\bf i}$ and $\dot{\bf q} \times {\bf q} = {\bf 0}$ (cf. Figure 2).
Type II Motions where $\theta = \pi$ and the axis of rotation is perpendicular to $\bomega$ . That is, $q_0 = 0$ and ${\bf q}\cdot\bomega = 0$ (cf. Figure 3).
Type III. Motions where $\theta = \theta(t)$ but ${\bf r}$ is neither normal to nor parallel to $\bomega$ . That is, $q_0 \ne 0$ and ${\bf q}\cdot\bomega \ne 0$ (cf. Figure 4).

Type I motions are prototypical constant angular velocity motions and, given the freedom to choose the reference basis $\left\{{\bf E}_1, {\bf E}_2, {\bf E}_3\right\}$ or, equivalently, ${\bf Q}_0$ , constant angular velocity motions can always be restricted to this type. For motions of this type ${\bf Q}(t_0) = {\bf I}$ at some instant $t = t_0$ and this guarantees that the constant $\bomega$ will be parallel to the axis of rotation ${\bf r}$ . Unfortunately, in many application areas, such as opthomology, the reference basis for ${\bf Q}$ is prescribed and Type II and Type III motions must be considered.

**Figure 2.** A representative pair of Type I motions and their representations. (a) The motions appear as straight lines on Steiner’s Roman surface, (b) the projection of the great circles on $S^3$ onto $\euclid^3$ appear as circles of unit radii, and (c) the corresponding spherical indicatrices of the instantaneous axis of rotation ${\bf i}$ (shown in blue), axis of rotation ${\bf r}$ (shown in red), and the vectors ${\bf e}_i$ . The motion of the rigid body corresponding to the Type I motion is best inferred from (c). That is, the rigid body appears to rotate about ${\bf i} \parallel \bomega$ . This figure is adopted from Novelia and O’Reilly [13].

**Figure 3.** A representative Type II motion and its representations. (a) The motion appears as a straight line on Steiner’s Roman surface, (b) the projection of the great circle on $S^3$ onto $\euclid^3$ appear as a circle of unit radius, and (c) the corresponding spherical indicatrices of the instantaneous axis of rotation ${\bf i}$ (shown in blue), axis of rotation ${\bf r}$ (shown in red), and the vectors ${\bf e}_i$ . The motion of the rigid body corresponding to the Type II motion is best inferred from (c) by noting that the body will appear to rotate about the instantaneous axis of rotation ${\bf i} \parallel \bomega$ (shown in blue). This figure is adopted from Novelia and O’Reilly [13].

**Figure 4.** A representative pair of Type III motions and their representations. (a) The motions appear as circles on Steiner’s Roman surface, (b) the projection of the great circles on $S^3$ onto $\euclid^3$ appear as circles with unit radii, and (c) the corresponding spherical indicatrices of the instantaneous axis of rotation ${\bf i}$ (shown in blue),the axis of rotation ${\bf r}$ (shown in red), and the vectors ${\bf e}_i$ . The reader should observe that although ${\bf e}_i$ rotate at constant speed about $\omega$ , the axis ${\bf r}$ and angle $\theta$ do not have constant rates of change. The motion of the rigid body corresponding to the Type III motion is best inferred from (c) by noting that the body will appear to rotate about the instantaneous axis of rotation ${\bf i} \parallel \bomega$ (shown in blue). This figure is adopted from Novelia and O’Reilly [13].

Geodesics of SO(3): alternate measures of distance on SO(3) inspired by axisymmetric and asymmetric rigid bodies

The kinematical line element or measure of distance for the configuration manifold in (17) is the simplest possible and is analogous to considering a spherically symmetric rigid body. However, inspired by the moment-free rotational motion of axisymmetric and asymmetric rigid bodies, alternative metrics are available. As we shall see, for the metrics considered here, Type I and Type II constant angular velocity motions are possible geodesics but motions with non-constant angular velocities are also possible.

In the case of an axisymmetric body, the body possesses two distinct principal moment of inertia $\lambda_i$ such that $\lambda_3 \ne \lambda_2 = \lambda_1$ . The rotational kinetic energy will change accordingly such that

(37) $\begin{equation*} T = \frac{1}{2} \bomega \cdot {\bf J} \bomega = \frac{1}{2} \lambda_3 \omega_3^2 + \frac{1}{2} \lambda_{\text{t}}\left( \omega_1^2 + \omega_2^2 \right), \end{equation*}$

where $\lambda_1 = \lambda_2 = \lambda_t$ . The balance of angular momentum for the rigid body is

(38) $\begin{equation*}\left[ \begin{array}{c}\lambda_t \dot{\omega}_1 \\\lambda_t\dot{\omega}_2 \\\lambda_3 \dot{\omega}_3 \\\end{array} \right] = \left[ \begin{array}{c}- \left( \lambda_3 - \lambda_2 \right) \omega_3\omega_2 = - \left( \lambda_3 - \lambda_{\text{t}} \right) \omega_3\omega_2 \\- \left( \lambda_1 - \lambda_3\right) \omega_3\omega_1 = \left( \lambda_3 - \lambda_{\text{t}} \right) \omega_3\omega_1\\- \left( \lambda_2 - \lambda_1\right) \omega_1\omega_2 = 0\\\end{array} \right]\end{equation*}$

A graphical representation of the solutions $\omega_i(t)$ to (38) is shown in Figure 9.2(a) of [28]. The solutions conserve the energy $T$ and the kinematical line-element for $SO(3)$ is

(39) $\begin{equation*} ds = \sqrt{\frac{2T}{m}} dt = \sqrt{\frac{1}{m} ( \lambda_t \omega_1^2 + \lambda_t \omega_2^2 + \lambda_3 \omega_3^2 )}dt. \end{equation*}$

We note that constant angular velocity motions are possible where $\bomega = \omega_{1_0} {\bf e}_1$ or $\bomega = \omega_{2_0} {\bf e}_2 + \omega_{3_0}{\bf e}_3$ where $\omega_{i_0}$ are constant. These constant angular velocity motions are geodesics of $SO(3)$ corresponding to Type I and Type II constant angular velocity motions (cf. Figures 2 and 3)). In addition, geodesics of $SO(3)$ also include a class of rotations where $\omega_1$ is constant but $\omega_{2,3}$ varying periodically in time. An example of one such motion is shown in Figure 5. For this rotational motion, the rigid body no longer executes a simple rotational motion: the third corotational basis vector $\bold{e}_3$ that is parallel to the axis of symmetry for the axisymmetric body traces out a circular path while $\bold{e}_1$ and $\bold{e}_2$ appears to be far more haphazard.

**Figure 5.** A representative motion of an axisymmetric body subject to an initial angular velocity $\bomega\left(t = 0\right) = \sum_{i}^{3} 10 \bold{E}_i$ . (a) The angle of rotation $\theta(t)$ given by the Euler parameter $q_0$ , (b) the instantaneous axis of rotation $\bold{i}$ (shown in blue) and the axis of rotation $\bold{r}$ (shown in red), (c) trajectories of the corotational basis vectors $\bold{e}_i(t)$ ( $\bold{e}_1$ (black), $\bold {e}_2$ (green), $\bold{e}_3$ (blue)). For the results shown in this figure $\lambda_1 = \lambda_2 = 22.083$ and $\lambda_3 = 3.333$ .

**Figure 5.** A representative motion of an axisymmetric body subject to an initial angular velocity $\bomega\left(t = 0\right) = \sum_{i}^{3} 10 \bold{E}_i$ . (a) The angle of rotation $\theta(t)$ given by the Euler parameter $q_0$ , (b) the instantaneous axis of rotation $\bold{i}$ (shown in blue) and the axis of rotation $\bold{r}$ (shown in red), (c) trajectories of the corotational basis vectors $\bold{e}_i(t)$ ( $\bold{e}_1$ (black), $\bold {e}_2$ (green), $\bold{e}_3$ (blue)). For the results shown in this figure $\lambda_1 = \lambda_2 = 22.083$ and $\lambda_3 = 3.333$ .

In the case of an asymmetric body, the body possesses three distinct principal moment of inertia $\lambda_k$ . The rotational kinetic energy has the representation:

(40) $\begin{equation*} T = \frac{1}{2} \bomega \cdot \bold{J} \bomega = \frac{1}{2} \lambda_1 \omega_1^2 + \frac{1}{2} \lambda_2 \omega_2^2 + \lambda_3 \omega_3^2. \end{equation*}$

The balance of angular momentum for the rigid body is

(41) $\begin{equation*}\left[ \begin{array}{c}\lambda_1 \dot{\omega}_1 \\\lambda_2\dot{\omega}_2 \\\lambda_3 \dot{\omega}_3 \\\end{array} \right] = \left[ \begin{array}{c}- \left( \lambda_3 - \lambda_2 \right) \omega_3\omega_2 \\- \left( \lambda_1 - \lambda_3\right) \omega_3\omega_1 \\- \left( \lambda_2 - \lambda_1\right) \omega_1\omega_2 \\\end{array} \right].\end{equation*}$

An illustration of the well-known graphical representation of the solutions $\omega_i(t)$ to (38) is shown in Figure 9.2(b) of [28]. The solutions conserve the energy $T$ and the kinematical line-element for $SO(3)$ is

(42) $\begin{equation*} ds = \sqrt{\frac{2T}{m}} dt = \sqrt{\frac{1}{m} ( \lambda_1 \omega_1^2 + \lambda_2 \omega_2^2 + \lambda_3 \omega_3^2 )}dt. \end{equation*}$

We note that constant angular velocity motions about a principal axis ${\bf e}_i$ are possible where $\bomega = \omega_{k_0} {\bf e}_k$ where $\omega_{k_0}$ is constant (e.g., $\bomega = \omega_{2_0}{\bf e}_2$ ). These constant angular velocity motions are geodesics of $SO(3)$ corresponding to Type I and Type II constant angular velocity motions (cf. Figures 2 and 3). In addition, geodesics of $SO(3)$ also include a class of rotations where $\omega_{1,2,3}$ varying periodically in time. An example of one such motion is shown in Figure 6. For this rotational motion, the rigid body no longer executes a simple rotational motion, rather it will appear to tumble in a haphazard manner. Other examples of a geodesic motion for this case can be found in Figures 9.3-9.5 of [28] and the pair of examples of a book tossed in the air and a tumbling T-handle in space (also known as the Dzhanibekov effect) discussed on this website.

**Figure 6.** A representative motion of an asymmetric body subject to initial angular velocity $\bomega_0 = \sum_{i}^{3} 10 \bold{E}_i$ . (a) The angle of rotation $\theta(t)$ given by the Euler parameter $q_0$ , (b) the instantaneous axis of rotation $\bold{i}$ (shown in blue) and the axis of rotation $\bold{r}$ (shown in red), and (c) trajectories of the corotational basis vectors $\bold{e}_i(t)$ ( $\bold{e}_1$ (black), $\bold {e}_2$ (green), $\bold{e}_3$ (blue)). For the results shown in this figure, $\lambda_1=27.083$ , $\lambda_2 = 22.083$ , and $\lambda_3 = 8.333$ .

**Figure 6.** A representative motion of an asymmetric body subject to initial angular velocity $\bomega_0 = \sum_{i}^{3} 10 \bold{E}_i$ . (a) The angle of rotation $\theta(t)$ given by the Euler parameter $q_0$ , (b) the instantaneous axis of rotation $\bold{i}$ (shown in blue) and the axis of rotation $\bold{r}$ (shown in red), and (c) trajectories of the corotational basis vectors $\bold{e}_i(t)$ ( $\bold{e}_1$ (black), $\bold {e}_2$ (green), $\bold{e}_3$ (blue)). For the results shown in this figure, $\lambda_1=27.083$ , $\lambda_2 = 22.083$ , and $\lambda_3 = 8.333$ .

For the results shown in Figures 5 and 6, initial conditions are $\phi(t=0) = \pi$ and $q_0 = 0$ , $q_1(t=0) = q_{1_0} = 0.20$ , $q_2(t=0) = \sqrt{1 - q^2_{1_0} } = 0.9798$ , and $q_3(t=0) = 0$ . Thus, with the help of (3) and (5),

(43) $\begin{equation*} {\bf r}\left(t = 0\right) = q_{1_0} {\bf E}_1 + \sqrt{1 - q^2_{1_0} } {\bf E}_2 \end{equation*}$

and the components $Q_{ik}\left( t= 0\right) = \left({\bf Q}\left( t= 0\right){\bf E}_k\right) \cdot{\bf E}_i$ of ${\bf Q}\left( t= 0\right)$ are

(44) $\begin{equation*} \mathsf{Q}\left( t= 0\right) = \left[ \begin{array}{c c c}2 q^2_{1_0} - 1 & 2 q_{1_0}\sqrt{1 - q^2_{1_0}} & 0 \\2 q_{1_0}\sqrt{1 - q^2_{1_0}} & 1 - 2 q^2_{1_0} & 0 \\0 & 0 & -1\end{array} \right]. \end{equation*}$

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