The hard turning process has been widely used in the field of hard material precision machining because of its high efficiency, low processing residual stress, and low environmental pollution. Due to its undesirably processing quality, it is still not a substitute for traditional grinding, so many studies have reported that the process has been optimized. However, there has been little research on the geometry optimization of hard cutting tools, which have a great influence on the traditional machining process. In this paper, two tools with different rake face shapes are designed. The finite element analysis method is used to compare their performance with a conventional plane tool while turning hardened steel. The results show that the cutting performance of the designed tool T1 and T2 (chip morphology, cutting force, and cutting temperature) and the quality of the machined surface are improved compared with the tool. The cutting force decreased by 12.72% and 14.74%, the cutting temperature decreased by 7.56% and 9.01%, respectively, and the surface residual stress decreased by 26.56% and 28.66%.

We show that the Fréchet distance between two piecewise linear surfaces can be decided in finite time, hence, the problem is decidable. For the special case that one of the surfaces is a triangle, we show that the problem is in PSPACE. In both cases, our computational model is a Turing Machine, and our algorithms rely on Canny's result [STOC 1988] that the existential theory of the real numbers is decidable in PSPACE.

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