Dualities and envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-spaces |
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Authors: | L Chen D H Pei M Takahashi |
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Institution: | 1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024 P.R. China;2. Muroran Institute of Technology, Muroran, 050-8585 Japan |
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Abstract: | 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. |
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Keywords: | de Sitter 2-space envelope frontal hyperbolic 2-space Legendrian duality |
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