Envelopes Of One-Parameter Families Of Spacelike Frontals In Pseudo-Spheres Space

In this paper,we focus on the differential geometry of envelopes of one-parameter families of spacelike frontals in pseudo-spheres space.Using the Legendrian dualities theory in hyperbolic and de Sitter 2-space,the moving frame are established along one-parameter families of spacelike Legendrian curves,the curvature of oneparameter families of spacelike Legendrian curves are introduced,and it is shown that the parallel curves of one-parameter families of spacelike Legendrian curves are the one-parameter families of spacelike Legendrian curves.It is also given the definition of envelopes of one-parameter families of frontals in hyperbolic and de Sitter2-space.In the process of studying the properties of envelopes,it was obtained that envelopes are Legendrian curves and the mapping are necessary and su cient condition for pre-envelope of one-parameter families of spacelike frontals,i.e.,envelope theorem,and the relationship between envelopes of parallel curves and envelopes of the original curve is proved.Using curvature and envelope theorems,we can show envelopes of the one-parameter families of frontals which are dual to each other are the same.Finally,two examples are used to explain the results of the theory.