Protecting Direction While Hiding Position: Scale-Invariant DP on the Quaternion Sphere

Position and direction carry different privacy value, yet trajectory differential privacy (DP) adds isotropic noise that destroys both at once. A published flight path exposes the operator's position, while the downstream tasks that consume these data, formation control and maneuver classification, depend on the direction. We decompose the trajectory into step lengths and unit headings, and add directional noise on the unit sphere: von Mises-Fisher on ${\mathbb{S}}^2$ for headings and Bingham on the quaternion sphere ${\mathbb{S}}^3$ for orientations. Direction-Aware Differential Privacy (DDP) obeys a closed-form $(\alpha ,\epsilon )$-R\'enyi DP bound that depends only on the concentration $\kappa$ and the angular sensitivity $\theta$, and never on the radial scale: directional privacy is scale-invariant. Under a mean-direction (vMF) or antipodal (Bingham) constraint, the noise is maximum-entropy. DDP is minimax optimal: it attains $\Theta (\theta /\sqrt{\epsilon })$ angular error and matches a minimax lower bound up to constant factors. Finally, a streaming variant (SDDP) releases each point as it arrives with an $O(1)$ privacy cost in the trajectory length and an $O(\sqrt{T})$ position drift. Because direction and step length are decoupled, DDP preserves the shape of a path while obscuring its exact coordinates, measuring privacy on the direction that downstream tasks consume.