Medical professionals have started favoring the use of non-contact intravenous injection robots owing to their importance during the COVID-19 outbreak. However, there are currently few studies considering the robot's needle insertion angle, and most of the needle insertion operations are performed at a steep angle. This increases the rate of puncture failure, and sometimes causes significant pain in patients depending on their individual differences. Therefore, the intravenous injection of the dorsal hand is performed in this study to investigate the determination of the robot's needle insertion angle, with a focus on the optimization of the measurement data to ensure accuracy in the calculation of the needle insertion angle. First, the space point cloud of the needle insertion area on the dorsal hand is obtained by combining a monocular camera with the linear structured light scanning method, and the dorsal hand plane is obtained via fitting dorsal hand point clouds using the least squares method. During the calibration process for the linear structured light system, the measurement error is eliminated by formulating an error function and using the optimization method to iteratively solve it. Subsequently, the needle insertion angle is determined based on the obtained needle insertion area plane. Finally, experiments are conducted for the accuracy verification of the proposed method. Based on the experimental results, the average error in the optimized structured light plane position is approximately 0.1 mm, and this serves as a foundation for subsequent automatic injection studies.