Abstract
As a one-parameter family of singular space curves, we consider a one-parameter family of framed curves in the Euclidean space. Then we define an envelope for a one-parameter family of framed curves and investigate properties of envelopes. Especially, we concentrate on one-parameter families of framed curves in the Euclidean 3-space. As applications, we give relations among envelopes of one-parameter families of framed space curves, one-parameter families of Legendre curves and one-parameter families of spherical Legendre curves, respectively.
Highlights
For regular plane curves, envelopes are classical objects in differential geometry [1,2,6,7,10,12]
Using the notion of stability for germs of family, they investigate the singularities of the envelope of the submanifolds in the Euclidean space via Legendrian singularity theory
If we look at the classical concept of envelope we see that the envelope is the set of characteristic points and avoiding singularities of the elements of the family
Summary
Envelopes are classical objects in differential geometry [1,2,6,7,10,12]. An envelope of a family of submanifolds in the Euclidean space were studied in [3,15]. Using the notion of stability for germs of family, they investigate the singularities of the envelope of the submanifolds in the Euclidean space via Legendrian singularity theory. By using implicit functions, the definition and calculation of the envelope of space curves are complicated. For singular space curves, the classical definitions of envelopes are vague. In [14], the second author clarified the definition of the envelope for families of singular plane curves. In [11], we clarified the
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