## 1 Introduction and Literature Review
The optimized system achieved a 6.6% increase in thrust, a 3.1% reduction in power consumption, and a 2.67% decrease in the maximum cavitation cavity size. These results demonstrate the method's strong potential for the design of high-efficiency propellers. However, while the inverse design and optimization method proposed by [PERSON] et al. [12] significantly enhances thrust and reduces cavitation, geometric deformation that occurs during casting or post-processing may prevent the final product from fully realizing the intended performance. The measurement system developed in this study, thus, serves as a geometric error verification platform that enables dimensional validation of optimized designs throughout the manufacturing stages. [PERSON] et al. [13] examined the feasibility and performance of additive manufacturing techniques and conventional manufacturing processes for the fabrication of underwater propeller blades. Various materials were tested and analyzed in their study. The findings indicated that although additive manufacturing remains limited in terms of the mechanical strength of its products, it excels in environmental resistance, manufacturing flexibility,and the ability to accommodate structural complexity. In particular, acrylonitrile butadiene styrene materials were demonstrated to have excellent corrosion resistance and molding efficiency, highlighting their potential to replace the metallic materials conventionally used in propeller manufacturing. In addition, [PERSON] et al. [14] proposed a multiobjective optimization method for propeller design to overcome the limitations associated with reliance on commercially available standard models. With the rapid advancement of 3D printing and CNC technologies, designers are increasingly creating customized and high-performance propellers. The study aimed to establish a viable alternative design framework that enables designers to effectively identify the optimal propeller configurations by balancing multiple performance criteria, thereby promoting design innovation and manufacturing flexibility. Although [PERSON] et al. [14] focused on small-scale propellers using evolutionary optimization, their work also involves geometric shape control and trade-offs among thrust, strength, and efficiency. The proposed measurement system in this study can be further extended as a feedback and rapid correction tool to verify and refine the geometric integrity of physical prototypes after optimization. From the initial wax injection to final surface finishing, the process used to cast a propeller typically involves 13 distinct stages. Throughout these stages, external mechanical stresses and temperature fluctuations can cause substantial dimensional and accuracy deviations in the cast components. These issues can be mitigated by improving the mold design concepts to reduce the effect of shrinkage. During the drying stage of lost-wax casting, deformation is likely to occur because the structure of the propeller blades is unsupported in the wax model. Thorough inspection of wax model deformation during manufacturing and the removal of components that exceed acceptable deformation thresholds help reduce the failure rate, thus enhancing the overall stability of propeller production. Currently, coordinate-measuring machines are used to verify whether the geometric features of propeller blades--such as their angle, diameter, and thickness--meet design specifications. However, because of the time-consuming nature of this inspection process and the high labor demand, only sampling inspections are feasible. For the handling of freeform surfaces and large-scale components, these methods are inefficient and difficult to automate. Therefore, a rapid inspection system is urgently needed to improve the inspection efficiency and overall product quality. In summary, despite advancements in casting and machining technologies, existing inspection systems are still labor-intensive, time-consuming, and inadequate for real-time quality monitoring of complex propeller geometries.