= Isotropic rotational diffusion in three dimensions
{title2=$\langle\mathbf u(t)\cdot\mathbf u(0)\rangle=e^{-2D_rt}$}
For <isotropic> turning in three dimensions, the tip of a marked <unit vector> $\mathbf u$ wanders over the unit <sphere>. Its <probability density function> $f$ obeys the orientational <Fokker-Planck equation>
$$
\frac{\partial f}{\partial t}=D_r\nabla_{\mathbf u}^2f,\qquad
C_1(t)=\langle\mathbf u(t)\cdot\mathbf u(0)\rangle=e^{-2D_rt},\qquad
\tau_1=\frac1{2D_r}.
$$
Here $\nabla_{\mathbf u}^2$ is the <Laplace-Beltrami operator> on the unit <sphere>. The factor two follows from the <sphere surface derivative identities>: $\nabla_{\mathbf u}^2u_i=-2u_i$, so each mean vector component decays at rate $2D_r$. This is a different orientation space from the circle of <planar rotational diffusion>, whose corresponding rate is $D_r$. For the same illustrative $D_r=0.25\,\mathrm{s^{-1}}$ and $t=2\,\mathrm s$, this three-dimensional <autocorrelation> is $e^{-1}\simeq0.368$.
One <unit vector> records a pointing direction, not every aspect of a <rigid body>'s <rigid body orientation>: rotation about that vector remains unspecified. Also, $C_1$ requires an identifiable arrow. An unmarked <ellipsoid>'s long axis alone does not supply one, and projecting a three-dimensional trajectory onto a camera plane does not automatically produce the planar model.
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