This paper presents a novel fourth-order formula for determining path intervals and comprehensively considers path interval formulas. In tool-path generation, a path interval is generally formulated as a scallopheight polynomial. Controlling scallop height in mechanical machining improves surface roughness or machining efficiency. We derived a novel fourth-order formula for determining path intervals after reviewing several formulas, then compared formulas. This clarified the differences between path interval formulas with graphic evidence. In micromechanical machining, an approximate expression has an advantage in computational cost but a disadvantage in accuracy. Although our proposed formula includes the fourth order-term scallop height, it requires low computational cost and can be applied to the determining path intervals for free-form surfaces in micromechanical machining. In addition, a correction method of the surface roughness on a free-form surface measured with a profilometer was proposed.

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