Segmental Geometry of Protrusive Trajectories

BACKGROUND: Mandibular protrusion is traditionally described using a single sagittal angular parameter, which does not capture directional variation along the protrusive trajectory. OBJECTIVE: To develop a rule-based geometric framework for analysing directional organisation of mandibular protrusion and characterising directional variation of protrusive trajectories. METHODS: Sixty-four protrusive trajectories were analysed using Global Protrusive Dynamics (GPD). Internal points of maximum directional deflection were identified within proportional search windows (25-50-25), defining four trajectory segments (S0-S4). Segmental directional angles were calculated for condylar and incisal trajectories. Robustness was assessed using independent procedures. RESULTS: Segmental analysis revealed non-uniform directional variation along the protrusive trajectory. Significant differences were observed between early (S1) and late (S4) trajectory segments, with large effect sizes. Median segmental directional angles showed progressive reduction along the condylar pathway. The incisal-condylar relationship also varied across segments. These patterns were consistent across workflows and stable under sensitivity analyses. CONCLUSION: Mandibular protrusion can be described as a structured spatial trajectory characterised by segment-dependent directional variation rather than a single sagittal inclination. The GPD framework provides a reproducible geometric approach for describing protrusive trajectory organisation and may support future longitudinal comparison of trajectory patterns under different functional conditions.