In recent years， rapid advancements in digital technology have positioned digital orthodontics as a critical research focus within the field of dentistry. Among the numerous challenges encountered during orthodontic treatment， designing an accurate dental arch line is fundamental for precisely calculating the target positions of teeth after treatment. The dental arch line should not only follow the natural growth patterns of the teeth but also satisfy aesthetic and functional requirements essential for optimal orthodontic outcomes. However， current automated tooth alignment methods typically model the dental arch line using Beta functions， which are inherently limited by their restricted degrees of freedom. This limitation often prevents these methods from generating curves that accurately capture the ideal dental arch form， especially when dealing with complex or irregular tooth arrangements. Moreover， orthodontists frequently require customized dental arch lines tailored to each patient’s unique oral condition. However， arch lines fitted solely from the patient’s initial intraoral scan may not always align with therapeutic or aesthetic expectations， necessitating labor-intensive manual adjustments. These challenges highlight the need for a flexible and precise approach to dental arch line design that effectively meets clinical standards and patient-specific requirements. Aiming to address these limitations， this paper proposes a novel dental arch line fitting method based on cumulative chord length parameterization combined with Hermite interpolation. This approach aims to enhance control over the dental arch shape， improve fitting accuracy， and provide orthodontists with a highly effective and efficient tool for designing and adjusting dental arch lines during orthodontic treatment planning.

The proposed method begins by inputting the patient’s intraoral scan data， which undergoes a series of preprocessing steps to ensure data quality and consistency. A tooth segmentation algorithm is then applied to accurately isolate each individual tooth， following internationally recognized dental segmentation standards. After segmentation， a landmark detection algorithm is employed to extract key landmarks from each tooth， capturing essential geometric and morphological features. These landmarks serve as the foundation for subsequent dental arch line fitting. Aiming to facilitate the interpolation process， the extracted landmarks are initially reparameterized using cumulative chord length parameterization. This process generates a naturally distributed set of interpolation points along the dental arch by accounting for the varying distances between adjacent landmarks， thereby preserving the true spatial relationships among teeth. Subsequently， Hermite interpolation is employed to construct the dental arch line through the parameterized points. By incorporating position and tangent information， Hermite interpolation enables the construction of smooth， continuous curves with enhanced local control. Aiming to ensure fitting accuracy and smoothness， a coefficient matrix is constructed to formulate a system of linear equations. Solving this system yields the final dental arch line， represented as a piecewise continuous function. This piecewise structure allows for precise local adjustments， making the method particularly effectively for accommodating complicated or irregular tooth arrangements. Furthermore， this paper introduces two new mathematical evaluation metrics： the mean shortest distance and the maximum shortest distance between the extracted landmarks and the fitted curve. These metrics offer an objective and robust means of assessing how accurately the generated dental arch line conforms to the patient’s actual dental morphology.

The proposed fitting method， which integrates cumulative chord length parameterization with Hermite interpolation， exhibits substantial improvements over traditional approaches in dental arch line fitting. First， compared to conventional Beta function-based methods， the proposed approach offers substantially greater flexibility by allowing the inclusion of additional control points. This increased degree of freedom directly addresses the limitations of Beta functions， particularly their inability to support localized shape modifications. The resulting dental arch line provides orthodontists with the flexibility to manually adjust specific， predefined control points， enabling localized adjustments tailored to individual patient needs. The proposed method excels in offering excellent controllability for global and local morphology adjustments of the dental arch line while maintaining high accuracy and smoothness across all regions， attributed to the use of its piecewise functional structure. Experimental evaluations further highlight the advantages of the proposed method. Qualitative analyses show that the generated curves more naturally align with actual dental arch shapes than those produced by conventional methods. Quantitative results， assessed using the proposed shortest distance-based evaluation metrics， confirm a notable improvement in fitting accuracy and alignment with natural tooth arrangements. Additionally， the proposed method enhances clinical flexibility， allowing orthodontists to efficiently adjust the dental arch line by manipulating a limited number of control points， minimizing the need for extensive manual corrections. In practical scenarios， the proposed fitting method is integrated into an existing automated tooth alignment system. This integration led to noticeably improved orthodontic outcomes， further validating the practical effectiveness and clinical applicability of the proposed method.

Compared to existing dental arch fitting methods， the proposed method based on cumulative chord length parameterization and Hermite interpolation demonstrates clear advantages in fitting accuracy and flexibility. This method effectively addresses key limitations of traditional approaches， such as difficulty in achieving an ideal dental arch line and limited adaptability to patient-specific variations. By notably increasing the degrees of freedom and enhancing the controllability of the fitting function， the method produces dental arch lines that are not only smooth and accurate but also highly customizable to meet the diverse clinical requirements of modern orthodontic practice. Furthermore， the introduction of quantitative evaluation metrics offers a systematic and objective framework for assessing fitting quality， ensuring that the resulting dental arch lines are aesthetically aligned and functionally sound. Beyond its technical advantages， the method also improves clinical efficiency by reducing the time and effort typically required for dental arch adjustments during treatment planning. Overall， the proposed method offers strong technical support for the advancement of digital orthodontics and holds substantial potential for broader clinical adoption. This paper establishes a solid foundation for further innovations in automated orthodontic treatment systems， opening new possibilities for personalized and precise dental care.