We consider envelopes of one‐parameter families of frontals in hyperbolic and de Sitter 2‐space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one‐parameter families of Legendrian curves by using the Legendrian dualities. Afterwards, we give definitions of envelopes for the one‐parameter families of frontals in hyperbolic and de Sitter 2‐space, respectively. We investigate properties of the envelopes. At last, we give relationships among those envelopes.

中文翻译：

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双曲和de Sitter 2空间中前参数一参数族的对偶性和包络

从对偶性的角度来看，我们分别考虑双曲和de Sitter 2空间中的一参数正面的包络。由于奇异曲线包络的经典概念不起作用，因此我们必须找到一种新的方法来定义双曲空间或de Sitter空间中奇异曲线的包络。为此，我们首先使用Legendrian对偶性介绍Legendrian曲线的单参数族的概念。然后，我们分别给出了双曲和de Sitter 2空间中一参数正面的信封的定义。我们调查信封的属性。最后，我们给出了这些信封之间的关系。