Significance analysis of scale factor in coordinate transformation under multi-condition simulation

As high-end precision equipment manufacturing advances toward the pursuit of extreme performance, precision measurement technology has increasingly become a critical driver in overcoming key technical bottlenecks. Serving as a vital mathematical bridge linking design models, machining data, and service, the accuracy of 3D coordinate transformation directly impacts the reliability of measurement results and their applicability in engineering practice. To systematically assess the accuracy of coordinate transformation models under varying measurement conditions, this study investigates two commonly used models — the six-parameter and seven-parameter transformations, differentiated by whether the scale factor is assumed to be 1. An experimental framework was constructed that encompassed four levels of error magnitude, four space sizes, and four point configurations, using these model types. Repeated simulations were conducted for each configuration. The statistical significance of the scale factor was evaluated through t-tests. Furthermore, the fitting capability and extrapolation accuracy of the models were analysed using the interior point (used for parameter solving) and external points (excluded from adjustment), respectively. Results indicate that error level is the dominant factor influencing transformation accuracy, while control point configuration has a significant impact under low-to-moderate error conditions. In contrast, the effects of the scale factor and spatial scale are relatively minor. In terms of model performance, the seven-parameter model demonstrates better fitting accuracy under high-error scenarios, whereas the six-parameter model exhibits superior prediction performance on external points. These findings provide quantitative insights for selecting appropriate transformation models and designing measurement systems, offering practical references for optimising coordinate transformation strategies in high-precision deformation monitoring and related applications.