# An Effective Algorithm for Finding Shortest Paths in Tubular Spaces
## 5 Computational Results
### Accuracy and Smoothness
As shown in Table 3, the proposed method always provides smaller \(RMSE\) and \(E_{max}\) than those obtained by [PERSON]'s algorithm for all of the tubes. Concretely, the average values among these tubes of \(RMSE\) and of \(E_{max}\) for the proposed solution are, respectively, 0.319% and 1.427% and approximately six times smaller than those given by [PERSON]'s algorithm (2.133% and 8.753%, respectively). As the path obtained by [PERSON]'s method must pass through nodes of the weighted graph, its position errors significantly depend on the meshing. These errors can be reduced if we increase the granularity of the mesh, but it will also increase the computation time. For the proposed method, the location of the obtained path is not forced to be the nodes of the graph that leads to smaller position errors. To extend the comparison results for different granularities of the meshed disks, Tube 6 (wave-shaped torus on a sphere) and Tube 7 (helical tubular surface) will be considered. The mesh is modified using different \(N_{\rho}\) and \(N_{\theta}\) as shown in Table 4. In these tests, the path length given by the proposed method are shorter than that proposed by [PERSON]'s algorithm for most cases. As a consequence, the proposed method also provides solutions with smaller position errors in comparison with [PERSON]'s algorithm. Even so, for the mesh with high granularity (\(N_{\rho}=20,N_{\theta}=256\)), [PERSON]'s algorithm can give a more \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & & & & & & & & & & \\ \hline Tube & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & Avg. \\ \hline [PERSON]’s algorithm & 0.506\% & 0.032\% & 1.922\% & 0.118\% & 0.007\% & 12.487\% & 1.655\% & 1.334\% & 1.140\% & 2.133\% \\ \hline Proposed algorithm & 0.002\% & 0.002\% & 0.004\% & 0.009\% & 0.001\% & 1.003\% & 0.510\% & 0.368\% & 0.974\% & 0.319\% \\ \hline \multicolumn{10}{c}{**Maximum Error**} \\ \hline Tube & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & Avg. \\ \hline [PERSON]’s algorithm & 4.343\% & 0.628\% & 11.812\% & 1.136\% & 0.226\% & 28.121\% & 8.556\% & 11.942\% & 12.017\% & 8.753\% \\ \hline Proposed algorithm & 0.012\% & 0.016\% & 0.026\% & 0.105\% & 0.015\% & 4.774\% & 2.077\% & 2.039\% & 3.782\% & 1.427\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Root mean square and maximum position errors of the ESP obtained by the two algorithm. The tube number is as given in Table 2.  
accurate result than that obtained by the proposed method but it requires a much longer computation time. The accuracy of [PERSON]'s solution increases with the increase in \(N_{\rho}\) and \(N_{\theta}\) while this is not always true with the proposed method. The reason is that a sparse grid can effectively constrain the movement of the VP in the correct direction in some cases. The proposed solution may become more wavy with a higher granular mesh. However, the length of the obtained paths only fluctuate in a narrow range (\(\pm\)0.5% for Tube 6) and (\(\pm\)0.1% for Tube 7). Eventually, when the number of nodes on the mesh is higher, the increasing rate in the computation time of [PERSON]'s algorithm is superior to that of the proposed method.