## 4 Propeller Blade Region Division  
First, define two region division parameters before propeller blade region division: minimum safety distance \(h_{safe}\), and maximum staggered ratio \(K_{max}\). The minimum safety distance refers to the minimum safety machining distance between the cutter tips on both sides during double-sided machining. The parameter size depends on the radius of the ball nose end mill in double-sided machining. The safety of double-sided machining can be ensured when the spacing between cutter tips on both sides is greater than or equal to the cutter radius; thus, the parameter size of the minimum safety distance is generally set to 1-1.5 times the radius \(r_{mill}\). The staggered ratio refers to the length proportion of the staggered machining tool path in a pair of symmetrical-tool-paths whose range is twice the range that does not meet the division parameter \(h_{safe}\) to the symmetrical-tool-paths, and the maximum staggered ratio \(K_{max}\) is the parameter that constrains the length proportion. To ensure that the propeller blades are machined symmetrically on both sides to the maximum extent, the maximum staggered ratio \(K_{max}\) is generally set to 0.1-0.2. Through two region division parameters, the blade surface is divided into four regions in this paper: tip region, internal region, variable region, and edge region. The division rules for the four regions are shown in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline
**Division Rules of \(K_{max}\)** & **Division Rules of \(h_{safe}\)** & **Corresponding Regions** \\ \hline staggered ratio \(\geq K_{max}\) & – & tip region \\  & the spacing between cutter tips on both sides \(\leq h_{safe}\) & edge region \\ staggered ratio \(<K_{max}\) & the spacing between cutter tips on both sides \(>h_{safe}\) & variable region \\  & (Path length equal to edge region) & variable region \\  & the spacing between cutter tips on both sides \(>h_{safe}\) & internal region \\ \hline \end{tabular}
\end{table}
Table 1: Rules for propeller blade region division.  
The tip region, which does not meet the division parameter \(K_{max}\), is generally thinner due to being away from the blade root. It is unable to guarantee the safety of double-sided symmetrical machining in this region; thus, single-sided alternating machining is implemented. Edge region, variable region, and internal region are obtained by dividing the machining-tool-paths meeting the division parameter \(K_{max}\). The edge region is the portion of each pair of symmetrical-tool-paths that does not meet the division parameter \(h_{safe}\). Due to problems such as insufficient spacing and the intersection of tool paths, it is also unable to guarantee the safety of double-sided symmetrical machining in this region; thus, double-sided staggered machining is used instead. The variable region is a transition region that connects the edge region and internal region, whose length is equal to the edge region. In this region, conversion between symmetrical machining and staggered machining is achieved by the interpolation of machining cutter points through the sigmoid deformation curve. The internal area meets the division parameter \(h_{safe}\) and implements double-sided symmetrical machining. The division of the propeller blade surface is shown in Figure 5.