Various touch-based interaction techniques have been developed to make interactions on mobile devices more effective, efficient, and intuitive. Finger orientation, especially, has attracted a lot of attentions since it intuitively brings three additional degrees of freedom (DOF) compared with two-dimensional (2D) touching points. The mapping of finger orientation can be classified as being either absolute or relative, suitable for different interaction applications. However, only absolute orientation has been explored in prior works. The relative angles can be calculated based on two estimated absolute orientations, although, a higher accuracy is expected by predicting relative rotation from input images directly. Consequently, in this paper, we propose to estimate complete 3D relative finger angles based on two fingerprint images, which incorporate more information with a higher image resolution than capacitive images. For algorithm training and evaluation, we constructed a dataset consisting of fingerprint images and their corresponding ground truth 3D relative finger rotation angles. Experimental results on this dataset revealed that our method outperforms previous approaches with absolute finger angle models. Further, extensive experiments were conducted to explore the impact of image resolutions, finger types, and rotation ranges on performance. A user study was also conducted to examine the efficiency and precision using 3D relative finger orientation in 3D object rotation task.