Our research establishes a rigorous mathematical framework for the six-degree-of-freedom (6-DOF) relative navigation of interconnected spacecraft using unit dual quaternions. Unlike traditional methods that decouple translational and rotational kinematics, often leading to singularities or inaccuracies in highly dynamic proximity operations, dual quaternions provide a unified, compact, and singularity-free representation of rigid body motion. By leveraging the Principle of Transference, we extended classical quaternionic rotation theory to encompass the full Euclidean group SE(3), treating the spacecraft's pose as a single algebraic entity. This approach inherently respects the natural coupling between orientation and position, which is critical for the precision required in docking and formation flying. Our project, focusing on the analytical derivation of relative motion equations without the use of stochastic filters, ensuring a deterministic foundation for state estimation, is structured around three primary tasks. Firstly, we developed the derivation of the dual quaternionic kinematic and dynamic equations of motion for a leader–follower spacecraft configuration. Secondly, we provided a proposal formulation of a closed-form analytical solution for relative pose estimation based on line-to-line and point-to-point correspondences. Finally, we developed a coordinate-invariant interpolation scheme for smooth trajectory generation between discrete states. Results demonstrate that the dual quaternion framework reduces the computational overhead associated with homogeneous transformation matrices by eliminating the need for frequent orthogonalization, replacing it with a simpler normalization process. Furthermore, the implementation in MATLAB for numerical validation and Python for high-fidelity visualization confirms that the proposed mathematical foundations provide superior accuracy in representing screw-based displacements, effectively mitigating gimbal lock and ensuring robust relative navigation for small satellite clusters.