In this paper, we derive the necessary and sufficient conditions of surface pencil pair interpolating Bertrand pair as common asymptotic curves in Galilean space G₃. Moreover, the conclusion to ruled surface pencil pair is also obtained. And the examples are given to confirm that the proposed methods are effective in product manufacturing by adjusting the shapes of the surface pencil pair.

中文翻译：

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将 Bertrand 对插值作为伽利略空间 G3 中公共渐近曲线的表面铅笔对

在本文中，我们导出了在伽利略空间G ₃中作为公共渐近曲线的表面铅笔对内插Bertrand对的充分必要条件。此外，还得到了直纹面铅笔对的结论。并通过实例验证了所提出的方法通过调整表面铅笔对的形状在产品制造中的有效性。