In high-energy physics research and industrial applications, measurement tasks in particle accelerator radiation zones are rendered extremely difficult due to radiation hazards, making direct human measurement infeasible. Additionally, since accelerators are typically installed in tunnels, space constraints and obstructions often cause many measurement target points to be hidden from view. To overcome these challenges, a rod-waving measurement technique based on laser trackers has been proposed to accurately measure the position of hidden points.The rod-waving measurement technique utilizes a laser tracker and a specially designed fiducial bar for hidden point measurement to perform spatial coordinate measurements. The fiducial bar for hidden point measurement is equipped with a target sphere. By waving the rod, the laser tracker acquires spatial coordinate data of the target sphere at different positions. Through spherical fitting of these data points, the three-dimensional coordinates of the hidden point can be determined. To address the issue of prolonged computation time associated with traditional spherical fitting algorithms, the algebraic spherical fitting algorithm was proposed. This new algorithm maintains accuracy while significantly reducing measurement time, thereby reducing the radiation exposure to measurement personnel. Subsequently, simulation experiments were conducted to test the impact of changing the length of the fiducial bar for hidden point measurement on fitting results, as well as the effects of varying zenith angles and spherical coverage ranges on the fitting results.The proposed algebraic spherical fitting algorithm requires only 18.85% of the computation time compared to the Gauss-Newton fitting method (Tab.1). In the simulation experiments, it was found that increasing the reference rod length from 0.5 meters to 3 meters resulted in a root mean square error (RMSE) of the fitting deviation changing by only 0.42 micrometers, indicating almost no impact on the fitting results (Tab.2). Additionally, it was discovered that under both global coverage (the horizontal angle is 0° to 360°) and hemispherical coverage (the horizontal angle is 180° to 360°) conditions, the RMSE of the fitting deviation remained consistently below 30 micrometers when the zenith angle was varied (Tab.4). Finally, practical experiments were conducted at the Beijing High Energy Photo Source laboratory using the AT960 laser tracker to measure four hidden points. The RMS of the fitting deviation for the measured data was 25.53 micrometers, meeting the precision requirement of being within 30 micrometers (Tab.6).The combination of laser trackers with the rod-waving method has demonstrated significant advantages in measuring hidden points within the radiation areas of particle accelerators. This approach overcomes spatial constraints and line-of-sight obstructions, offering extremely high measurement accuracy and reliability. The proposed algebraic spherical fitting algorithm significantly outperforms the commonly used Gauss-Newton method in terms of computation time, requiring only 18.85% of the latter's time. This not only enhances measurement efficiency but also substantially reduces the duration of radiation exposure for operators. Simulation experiments revealed that the algorithm is highly stable against changes in the fiducial bar length, with the root mean square error (RMSE) of the fitting deviation changing by only 0.42 micrometers. The experiments also showed that whether under global or hemispherical coverage, the RMS of the fitting deviation remained consistently below 30 micrometers when the zenith angle was varied, demonstrating the algorithm's reliability. Ultimately, through field measurements, an RMS of the fitting deviation of 25.53 micrometers was achieved, successfully meeting the precision requirement of being within 30 micrometers. This provides an efficient and reliable solution for precise hidden point measurement in special environments such as particle accelerator radiation areas.