0

Let $F(x, y, z, \lambda)=0$ be a parametric family of implicit surfaces. Sometimes the envelope of the family exists as another surface, but at other times it may degenerate to a curve or a point, or even it can be empty.

Is there any know condition that guaranteed the existence of the envelope as a surface?

Bumblebee
  • 18,974
  • 1
    You did not search this site very well. Here are various links to my own answers (some in the case of curves, but you can easily generalize): here, here, here. – Ted Shifrin Aug 13 '23 at 20:53
  • @TedShifrin: Thank you for your comment. What would be a good book to learn this stuff? None of the undergraduate DG books that I found does not discuss this topic. – Bumblebee Aug 14 '23 at 04:07
  • 1
    I don’t know any modern DG books that treat this topic. The implicit function theorem is the necessary tool. The result in my posts appears as an exercise in my Multivariable Mathematics text. – Ted Shifrin Aug 14 '23 at 04:25
  • @TedShifrin: I see. It is on page number 261. Yet it is unclear to me how this generalizes to a family of surfaces as the analogous matrix is not a square matrix. Do you mind writing a complete solution to this? It would be beneficial for lots of others as well. – Bumblebee Aug 14 '23 at 12:30
  • Sorry. I didn’t pay enough attention. I was expecting two parameters. What even is the definition if the envelope in your case? – Ted Shifrin Aug 14 '23 at 16:27
  • @TedShifrin: "locus of characteristic curves" – Bumblebee Aug 14 '23 at 16:40
  • Please edit your post to include a detailed definition. This is not something I'm familiar with outside of PDE. – Ted Shifrin Aug 14 '23 at 16:46

0 Answers0