Envelopes - notion and definiteness

We examine critically some of the existing descriptions of the envelope of a 1-parameter family of surfaces in 3-space. An old, natural, description is that the envelope is the union of the characteristic curves (defined as the “limit intersection curves” of surfaces from the family). This description fell in disrepute during the 20th century. We claim that this relegation of the limit intersection curves made some other aspect of the theory less justified, namely the definite article “the” in the phrase “the envelope”. We intend to re-establish the limit intersection curves on a rigourous basis, – thereby also re-establishing the old obvious argument for the definiteness of envelopes. The content of the present note was presented at a talk at the Logic Year at the Mittag-Leffler Institute in Stockholm in April 2001. The key formula (1) was found during discussions with Gonzalo Reyes.