Recognition of Computationally Constructed Loci

摘要

We propose an algorithm for automated recognition of computationally constructed curves and discuss several aspects of the recognition problem. Recognizing loci means determining a single implicit polynomial equation and geometric invariants, characterizing an algebraic curve which is given by a discrete set of sample points. Starting with these discrete samples, arising for example from a geometric ruler and compass construction, an eigenvalue analysis of a matrix derived from the data leads to proposed curve parameters. Utilizing the construction itself, with its free and dependent geometric elements, further specifications of the type of constructed curves under genericity assumptions are made. This is done by a second eigenvalue analysis of parameters of several generically generated curves.