## 3 Results
### Verification on Vaginal Examination Simulation  
We have also applied the proposed framework to our VE training simulation, as shown in Figure 1, and measured the latency and accuracy. The cervix model and vulva model are composed of 2160 and 2400 vertices, respectively, with a total of 32 (4 \(\times\) 4 \(\times\) 2) and 72 (6 \(\times\) 6 \(\times\) 2) control points (Figure 1). In the simulation, as the training progresses, the position and orientation of the baby's head, as well as the cervix model, change and result in the entrance of the cervix model being dilated. Therefore, simulation states are defined as the cervix diameter, position of the cervix and baby's head, and orientation of the baby's head. However, to simplify the latency and accuracy measurements, we used only the cervix diameter as a simulation state. Two shapes were predefined to simulate the dilation of the cervix model: At Rest and Dilated. The global deformation was computed by blending the two shapes. The deformation from At Rest to Dilated was divided into 50 steps. The inverse matrices of 50 blended matrices were precomputed and stored in the server beforehand to reduce the computational burden during a simulation. Mapping matrices of the At Rest and Dilated shapes were stored in the client, and a blended matrix was computed based on Equation (3) when a weighting value was received. The diameter of the cervix was determined from the simulation state. Control points were computed by the corresponding precomputed inverse matrix and then transmitted to the client, together with the corresponding weighting value. A total of 4560 pairs of vertices and normals were needed to directly render a virtual scene on the client. A total of 9120 vectors were sent to the client for 500 steps. The average and maximum latency caused by sending 9120 vectors were 18,969 and 40,092 ms, respectively. In the case of the proposed framework, 104 control points were needed to render the virtual scene. The average and maximum latency over 500 simulation time steps caused by sending 104 control points were 7 and 34 ms, respectively. The average and maximum latency are summarised in Table 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \begin{tabular}{c} **The** \\ **Information** \\ \end{tabular} & \begin{tabular}{c} **Number of** \\ **Vectors** \\ \end{tabular} & \
\begin{tabular}{c} **Average** \\ **Latency** \\ \end{tabular} &
\begin{tabular}{c} **Maximum** \\ **Latency** \\ \end{tabular} \\ \hline w/o the proposed & 4560 surface vertices & 9160 & 18,969 ms & 40,092 ms \\ framework & 4560 normals & & & \\ \hline w/the proposed & 104 control points & & & \\ framework & weight value & 104 & 7 ms & 33 ms \\ \hline Reduction rate & & 93.0\% & 99.9\% & 99.9\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Average and maximum latency comparison and reduction rate in the VE training simulation.  
Figure 11: Root mean square position error of the Stanford Bunny model over simulation time steps.  
The cervix model with large deformation was used for the accuracy analysis. Figure 12 shows the RMS position error as a function of the simulation time steps. The error increases by up to about 0.5 mm when the cervix model dilates and contracts. The error increases by up to 2 and 2.8 mm when the user grabs and manipulates some points of the dilated and initial cervix model, respectively. The error is proportional to the amount of manipulation by the user. However, the error is relatively small when the cervix model dilates and contracts since the global deformation is transferred by a blended mapping matrix. The diameter of the cervix model is 100 mm; therefore, the maximum relative error is less than 3%.