Envelopes of one-parameter families of framed curves in the Euclidean space

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.