Objectives: Accurate assessment of orthognathic surgical accuracy is essential in the evaluation of operative techniques. Surgical accuracy is often reported as rotational and translational deviations from planned positions. This results in 6 separate values, translation in three planes, anterior-posterior (AP), superior-inferior (SI) and medial-lateral (ML) and rotations about three axes, pitch, roll and yaw. However, rotations will influence 3-dimensional positions and translational discrepancies. Methods: We have derived a mathematical formula using Euclidean geometry and quadratic functions that quantifies the impact of rotations on translational discrepancies. This allows for the calculation of a total discrepancy value that incorporates the three translations and rotations. Furthermore, we developed an interactive web-based application using the open-source shiny R package. Results: We have successfully reduced equations from Euclidean geometry into a quadratic form. The equation is as follows, [4(sinθ/2)2-2]x2 + [8d(sinθ/2)2-2d]x + 4d2(sinθ/2)2 = 0, where θ represents the rotational discrepancy in radians and d represents the translation discrepancy. This allows us to solve for the correction needed to be made to translational discrepancies to account for the influence of rotational discrepancies. We successfully developed a web application with a user-friendly graphical user interface. Clinicians upload their own data in the excel (.xlsx) file format and the application automatically performs the necessary calculations over many patients, returning a downloadable table of results. Conclusion: We present a mathematical formula incorporated into a web-application to combine translational and rotational discrepancies for deeper insight when evaluating orthognathic surgical accuracy. Clinical Relevance: This allows surgeons to account for rotational influence on 3-dimensional translational discrepancies.

The authors have declared no competing interest.

This study did not receive any funding.

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