# Hermite Quartic Splines for Smoothing and Sampling a Roughing Curvilinear Spiral Toolpath
## 1 Introduction
Since 'polynomial' interpolators are not yet widely adopted in the industry, efforts are needed to smooth the toolpath and improve its degree of continuity. Figure 1: Interpolation between two structure curves with a Fresnel vector. The simplest method to achieve this objective is the use of a Finite Impulse Response filter (**FIR**); [21] suggests a moving average technique. However, the use of such filters may result in a loss of tool position control, potentially leading to a machining in inappropriate zones. Other various studies [22; 26; 27] achieve three of five-axis trajectory smoothing, via B-splines or Bezier curves. Table 1 shows a summary of the methods proposed in the literature summarizing their main focuses. This article proposes a method to reparameterize the structure curves of the curvilinear spiral with a set of Hermit quartic spline patches \(C^{2}\) connected (continuity in tangency and curvature). Analytically, the spiral-like toolpath, formed through linear combination of these curves, maintains the \(C^{2}\) continuity, ensuring a smaller machining time. Moreover, the proposed method eliminates the need of Fresnel's vector to establish the spiral parametrization, so the toolpath can be directly created without partitioning the \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Years** & **Authors** & **Main Focuses** & **Benefits and Limitations** \\ \hline \textless{}2000 & \multirow{2}{*}{Held} & Generation of toolpaths on arbitrary geometry & \multirow{2}{*}{/} \\  & & Zig-zag or parallel contour & \\ \hline
2003 & \multirow{2}{*}{Bieterman [21]} & Spiral toolpath & \multirow{2}{*}{
\begin{tabular}{l} Suitable only for globally convex pocket \\ Not suitable for pocket with island \\ \end{tabular} } \\  & & PDE approach & \\ \hline
2004 & \multirow{2}{*}{[PERSON] [28]} & Corner smoothing & Local smoothing \\ \hline
2009 & \multirow{2}{*}{[PERSON] [29]} & Spiral toolpath & \multirow{2}{*}{
\begin{tabular}{l} Use of Voronoi diagrams \\ Not suitable for island \\ \end{tabular} } \\  & & Geometric approach & \\ \hline
2011 & \multirow{2}{*}{[PERSON] [18]} & Spiral toolpath & \multirow{2}{*}{
\begin{tabular}{l} No smoothing \\ Suitabel for pockets with island but with \\ composite spiral \\ \end{tabular} } \\  & & & \\ \hline
2011 & \multirow{2}{*}{[PERSON] [30]} & Spiral toolpath & \multirow{2}{*}{
\begin{tabular}{l} Only globally convex pocket \\ Suitabel for pocket with a centered island \\ \(G^{1}\) toolpath \\ \end{tabular} } \\  & & & \\ \hline
2013 & \multirow{2}{*}{Xu [23]} & Spiral toolpath & \multirow{2}{*}{
\begin{tabular}{l} Not suitable for pocket with island \\ No smoothing \\ \end{tabular} } \\  & & Geometric approach & \\ \hline
2018 & \multirow{2}{*}{Held [19]} & Spiral toolpath & \multirow{2}{*}{
\begin{tabular}{l} Simplier to implement \\ Polygonal spiral (\(G^{0}\)) \\ \end{tabular} } \\  & & Geometric approach & \\ \hline
2023 & \multirow{2}{*}{Hua [27]} & Curve smoothing & \multirow{2}{*}{Not integrated in a spiral construction} \\  & & B-Spline & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Research timeline. Figure 2: Partition of a pocket [25]. pocket. It allows a generalization of Bieterman's procedure for any \(2.5D\) pocket geometries, even with non-convex geometry.