Abstract Introduced are special evolutoid curves and spatial involutoid curves based on the special evolutoid curves. The special evolutoid is the striction curve to the family of constant pressure angle contact normals defined by a single transverse curve. Each axial position between the toe and heel of mating hyperboloidal pitch surfaces defines a unique transverse curve, and consequently, a special evolutoid and its spatial involutoid. The active tooth flank is comprised of a family of involutoid segments. This family of involutoid segments is used to define conjugate tooth flanks for hypoid gear sets. For hypoid gear sets, the pressure angle varies axially with each transverse curve. The generation of spatial involutoid curves is similar to the classical and well understood case of planar involute curves; the involutoid curve is obtained by the unwrapping of a taut chord from a planar and circular evolutoid. The line of action between the two conjugate flanks is simultaneously coincident with the input and output gear evolutoids. This process differs from the generation of planar involute curves in that the spatial involutoid curves are non-planar and there exists slip as the taut chord is unwrapped. Three gear sets are presented to showcase the process.

中文翻译：

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用渐开线和空间渐开线来定义准双曲面几何

摘要 介绍了特殊演化曲线和基于特殊演化曲线的空间渐开曲线。特殊的演化曲线是由单一横向曲线定义的恒压角接触法线族的收缩曲线。配对双曲面节距表面的趾部和跟部之间的每个轴向位置都定义了一条独特的横向曲线，因此，形成了一个特殊的渐开线及其空间渐开线。活动齿面由一系列渐开线段组成。该系列渐开线段用于定义准双曲面齿轮组的共轭齿面。对于准双曲面齿轮组，压力角随每条横向曲线轴向变化。空间渐开线曲线的生成类似于平面渐开线曲线的经典且广为人知的情况；渐开线曲线是通过从平面和圆形渐开线展开拉紧弦获得的。两个共轭齿面之间的作用线同时与输入和输出齿轮渐开线重合。该过程不同于平面渐开线曲线的生成，因为空间渐开线曲线是非平面的，并且在拉紧弦展开时存在滑移。展示了三个齿轮组以展示该过程。