# An Open-Source Algorithm for 3D ROck Slope Kinematic Analysis (ROKA)
## 4 Discussion  
The comparison between the results of the traditional kinematic analysis (Markland test) and the results of the ROKA analysis performed on two different rock slopes highlights the main advantages of the last methods. The first is the strong reduction of the discontinuity intersections that must be verified. While traditional methods calculate all the possible intersections between the supposed infinite planes of discontinuity, the ROKA algorithm calculates only the \"real\" discontinuity \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Discontinuity Set** &
\begin{tabular}{c} **Considered** \\ **Discontinuities** \\ \end{tabular} & **Planar Failure** & **Flexural Toppling** \\ \hline Bedding & 277 & 0 & 0\% & 3 & 1\% \\ K2 & 436 & 123 & 28\% & 29 & 7\% \\ K3 & 167 & 11 & 7\% & 21 & 13\% \\ Random set & 183 & 0 & 0\% & 0 & 0\% \\ All & 1063 & 134 & 13\% & 53 & 5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 8: According to the ROKA Kinematic Analysis, the table shows the number of discontinuities that could activate planar and flexural toppling modes of failure. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Total Number of Critical** & **Wedge Failure** & **Direct Toppling** \\ \hline
4667 & 538 & 710 \\
**Percental** & 12\% & 15\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: According to the ROKA Kinematic Analysis, the table shows the number of discontinuities intersections that could activate wedge and direct toppling modes of failure. Figure 20: Rendering of the 3D model indicating the results of the ROKA Kinematic Analysis of the possible modes of failure. The color scale represents the critical value of discontinuity intersections that could activate (**a**) wedge failure and (**b**) direct toppling, (see Section 2.5 for the critical value definition). intersections, considering not only their orientation but also their position and dimension as mapped onto the DOM. In the two examined study cases, this reduction is particularly evident; from 1,838,093 and 564,373 to 4349 and 4667 intersections for cases A and B, respectively, with a difference of more than 99%. Consequently, the ROKA algorithm detects a lower number of critical discontinuity intersections with respect to the traditional KA (Table 10). The second main advantage of the ROKA method is the more accurate determination of the different local slope orientations using a movable scan-sphere, the radius of which can be defined by the operator. On the contrary, the traditional KA generally considers only one orientation, i.e., the slope's mean orientation (Figure 21). This different geometrical modeling of the slope influences the results of the stability analysis. The differences in the number of discontinuities that can cause planar failure and flexural toppling for the study cases A and B are synthesized in Table 11.  
The traditional KA tends to underestimate the number of critical discontinuity planes and the number of discontinuity sets because the different slope orientations are not considered (see Tables 2, 4, 6 and 8). For example, in the case A, whereas the traditional \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case of Study**} & \multirow{2}{*}{**Type of KA**} & \multicolumn{2}{c}{**Considered Discontinuities**} & \multicolumn{2}{c}{**Wedge Failure**} & \multicolumn{2}{c}{**Direct Toppling**} \\  & & **Intersections** & & & & \\ \hline \multirow{2}{*}{Case A} & Traditional & 1,838,093 & 202,196 & 11\% & 74,411 & 4\% \\  & ROKA & 4349 & 1214 & 28\% & 153 & 4\% \\ \multirow{2}{*}{Case B} & Traditional & 564,373 & 73,536 & 13\% & 14,007 & 2\% \\  & ROKA & 4667 & 538 & 12\% & 710 & 15\% \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison of the results of the different KA of wedge and direct toppling modes of failure.