Figure 2.

The frame configurations and reduction process were presented with the pelvic 3D printed and reconstruction models. (a) The frame configurations include two larger side arc bars (1), which are used for securing the intact hemipelvis, and two smaller side arc bars (2), which are used for connecting and holding the injured hemipelvis. The injured hemipelvis can be rotated or translated by the smaller side arc bars through the LC II screw (3) in a controlled manner. The rotations of hemipelvis can be performed around the center of the anterior arc on the anterior arc bar (4) in the LC II plane and the LC II screw in the plane vertical to LC II screw, and along the smaller side arc bars in the sagittal plane. (b) The reduction process of the rotational displacements of the injured hemipelvis can be explained by four hemipelves in four different virtual places. Hemipelves (1) and (2) are the hemipelves at displaced and reduction places, respectively. Hemipelves (3) and (4) are two intermediate places, which are rotated from hemipelves (1) and (2), respectively. The rotation degrees from hemipelvis (1) to (3) and hemipelvis (2) to (4) can be calculated by using the intersection degrees of their respective LC II screws in the sagittal and LC II planes, respectively. In this condition, there will be only one self-rotation around the LC II screw (5) with hemipelves (3) or (4), which are parallel to each other in space and can be calculated using the matrix transformation of 3D rotation around an arbitrary axis as follows:$\left[\begin{array}{ccc}{n}_x^{\mathrm{2}}(\mathrm{1}-\cos \theta )+\cos \theta & n_y{n}_x\left(\mathrm{1}-\cos \theta \right)+n_z\sin \theta & n_x{n}_z\left(\mathrm{1}-\cos \theta \right)-n_y\sin \theta \\ n_y{n}_x\left(\mathrm{1}-\cos \theta \right)-n_z\sin \theta & n_y^{\mathrm{2}}(\mathrm{1}-\cos \theta )+\cos \theta & n_z{n}_x\left(\mathrm{1}-\cos \theta \right)+n_x\sin \theta \\ n_z{n}_x\left(\mathrm{1}-\cos \theta \right)+n_y\sin \theta & n_z{n}_y\left(\mathrm{1}-\cos \theta \right)-n_x\sin \theta & n_z^{\mathrm{2}}(\mathrm{1}-\cos \theta )+\cos \theta \end{array}\right]$. (c) During the operation, the self-rotation around the LC II screw could be completed with a specialized protractor.