Envelope computation in dynamic geometry systems

Considerable attention has been given to the computation of geometric loci in dynamic geometry, from both graphical and equational viewpoints. Recent work has established a rigourous approach to the last issue, solving the subject for the algebraic realm. Nevertheless, although the automatic computation of envelopes could be seen as an analogous problem, there are cases where unexpected difficulties emerge. In this talk we review the state of the art of common dynamic geometry software when dealing with envelopes. Despite their maturity in other subjects, envelopes are generally considered as purely graphic objects in such environments, without any analytic knowledge about them. We also describe a data structure needed to cope with envelopes in dynamic geometry, and a Sage program able to compute usual envelopes in an efficient way. Finally, in order to show the complexities of such computations, we discuss the envelope of a simple family of ellipses. The search for such envelope will illustrate two facts: 1. There is no general agreement between dynamic geometry developers about the definition of envelope, and 2. Currently, the simple application of computer algebra techniques is not enough to automatically solve the problem.