# Hermite Quartic Splines for Smoothing and Sampling a Roughing Curvilinear Spiral Toolpath
## 6 Conclusions  
This article proposes a new method to construct curvilinear spiral toolpaths from structure curves obtained by the FEM solution of PDE inspired from a thermal problem. To reduce the tangency discontinuities due to discretization, the structure curves of the spiral are rebuilt with Hermit quartic spline patches. This method performs the following:  
* Models structure curves for any geometry of pockets thanks to the associated thermal model;
* Guarantees a \(C^{2}\) continuity of the structure curves and enhances the continuity level of the curvilinear spiral;
* Creates a toolpath having, globally, a low curvature;
* Allows the user to define a maximum chordal error along the structure curves.  
Two representative geometries of pockets are used as examples and show reductions of 10 to 18% of machining time as compared to the state of the art. The proposed method also offers a generalization to non-convex pockets as compared to methods from the literature. In the future, the goal will be the use of Hermite quartic curves directly in a numerical controller that can handle spline interpolations.  
Conceptualization, C.L.; methodology, C.L.; software, C.L. and S.L.; validation, C.L.; formal analysis, C.L. and E.R.-L.; investigation, C.L.; writing--original draft preparation, C.L.; writing--review and editing, C.L., S.L. and E.R.-L.; visualization, C.L.; supervision, E.R.-L.; project administration, E.R.-L.; funding acquisition, E.R.-L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Institutional Review Board Statement**: Not applicable.  
**Informed Consent Statement**: Not applicable.  
**Data Availability Statement**: The data presented in this study are available on request from the corresponding author.  
**Acknowledgments:** We wish to thank the LURPA lab from ENS Paris-Cachan for the collaboration proposition and the VPOp licensing agreement, [PERSON] for the developed software and [PERSON] for its parametrization, [PERSON] for the language correction of the paper.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Toolpath Location** & **Fitting Method** & **Milling Time [s]** & **At / [\%]** \\ \hline Internal (Iso14) & RAW & 9 & / \\  & HQS & 7.3 & \(-19\) \\ \hline Intermediate (Iso10) & RAW & 15.2 & / \\  & HQS & 12.2 & \(-20\) \\ \hline External (Iso3) & RAW & 22.3 & / \\  & HQS & 17 & \(-24\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Simulated milling time comparison for 3 structure curves.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Curvature Absolute Value (\(\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ }}}}}}}}}}}}}\)} \\ \hline \hline \multirow{3}{*}{\
\begin{tabular}{c} Cu95 \\ Cu99 \\ \end{tabular} } & \multicolumn{6}{c}{Entire Spiral} & \\  & & & 0.12 & & \\  & & & & 0.72 & & \\ \hline \multirow{3}{*}{
\begin{tabular}{c} Cu95 \\ Cu99 \\ \end{tabular} } & Iso3 & \multicolumn{2}{c}{Iso10} & Iso14 & \\  & RAW & HQS & RAW & HQS & RAW & HQS \\ \hline Cut95 & 0.38 & 0.07 & 0.45 & 0.07 & 0.44 & 0.10 \\ Cut99 & 1.07 & 0.08 & 1.05 & 0.13 & 1.51 & 0.12 \\ \hline \hline \end{tabular}
\end{table}
Table 5: HUG pocket—curvature quality statistical indicators.