Hyperplane Envelopes and the Clairaut Equation

The subject of envelopes has been part of differential geometry from the beginning. This paper brings a modern perspective to the classical problem of envelopes of families of affine hyperplanes. In the process, the classical results are generalized and unified. A key step in this process is the use of “generalized immersions”. These have been described elsewhere but, briefly, every classical immersion defines a generalized immersion in a canonical way so that generalized immersions can be understood as ordinary immersions “with singularities.” Next, the concept of an envelope is given a modern definition, namely, an envelope is a generalized immersion solving the family that has a universal mapping property relative to all other full rank solutions. The beauty of this approach becomes apparent in the “Envelope Theorem”. With one mild assumption, namely that the associated family of linear hyperplanes is immersed, it is proven that a family of affine hyperplanes always has an envelope, and that envelope is essentially unique.