# Hermite Quartic Splines for Smoothing and Sampling a Roughing Curvilinear Spiral Toolpath
## 1 Introduction  
Pocketing \(2.5D\) cavities, which is common from an industrial point of view (e.g., in aeronautical structure parts or aerospace electronic carnkcase), is a time-consuming operation. As shown by [1], manufacturing costs are time dependent and typically represent, for dies and molds, 30% of the total price of a piece [2]. In this context, many studies aim to reduce the machining time. The main lines for optimization are enhancing the understanding of cutting phenomena [3; 4; 5], improving tool performance [6], selecting the best tools, defining the optimal milling parameters [7; 8; 9], and, finally, enhancing the toolpath geometry [10; 11; 12]. From this last perspective, the most common approach is to work on the tool trajectory to increase its speed. Multiple studies (e.g., [13; 14]) have demonstrated that tangency discontinuities along the trajectory result in increased machining time. These discontinuities necessitate changes in the velocity vector (both in direction and magnitude) along the trajectory, which challenges the machine tool's acceleration and jerk capabilities. As these capacities are limited, the numerical control system of the machine imposes restrictions on these variations, so the tool requires additional time and distance to achieve the programmed feed rate after negotiating corners or halts along the toolpath. Therefore, it is essential to minimize such discontinuities, for which two methods exist:* Local adjustments on classical toolpath (Zig-Zag or Contour Parallel) are employed to eliminate or to smooth out the discontinuities [15; 16; 17];
* Global strategy to build a spiral-like trajectory (or curvilinear spiral). Macroscopically, this kind of toolpath begins at a center point of the pocket and expands outward without \"stop and go\". The toolpath morphs progressively to the shape of the pocket boundary. Natively, curvilinear spirals (or spiral-like toolpath) have a higher degree of continuity [18; 19; 20; 21]. Primarily, two approaches are used to build curvilinear spirals:
* Geometrical approaches such as mapping [22; 23], morphing [24], or the medial axis method [19; 20];
* Differential approaches using an elliptical partial differential equation (**PDE**) [21] or a level-set propagation model [18]. In those cases, curvilinear trajectories are generated through interpolation between selected structure curves, which are the solutions of the used PDE. To determine a point on a curvilinear path, an interpolation method is needed. [PERSON] et al. [21] use the intersection of a Fresnel vector (winding angle) and two discrete structure curves (Figure 1). The trajectory point is established through a linear angular interpolation between the two intersecting points (Equation (1)):  
\[\rho_{S}(\theta_{i})=\rho_{int}(\theta_{i})+\big{(}\rho_{ext}(\theta_{i})- \rho_{int}(\theta_{i})\big{)}.\frac{\theta_{i}}{2\pi}\,\ \theta_{i}\in[0,2\pi[ \tag{1}\]  
with \(\rho_{S}(\theta_{i})\), discrete polar equation of a spiral-like turn; \(\rho_{ext}(\theta_{i})\), discrete polar equation of the external structure curve; \(\rho_{int}(\theta_{i})\), discrete polar equation of the internal structure curve. This method is effective when the cavity is sufficiently convex (i.e., without significant protrusions; mathematically, when the polar function of the pocket border is explicit). However, for pockets with more complex geometries, a partitioning task is required as proposed by [25] and illustrated in Figure 2.  
Moreover, the structure curves are constructed through interpolations between a series of points calculated using finite element method software. In the context of G1 linear interpolation, the trajectory consists of numerous small segments. Although the trajectory may exhibit characteristics adapted to high feed rates on a macroscopic scale, the tangency discontinuities inherent to this method significantly increase the machining time on a smaller scale.