Skirting subsets of the plane, with application to marginal stability curves

Abstract

An easily implemented algorithm is described for tracing the margin of a plane region defined by a predicate. Given a point inside and one outside, a sequence of marginal points is produced. The algorithm is a modified specialization of the `simplicial decomposition' method for n equations in n+1 dimensions. The case n=1 has special properties and its importance motivates their present exploitation. It is directly applicable to finding level curves. It does not require differentiability and copes well with cusps. Two questions of accuracy are the proximity of the outputs to the margin and the proximity of the margin to the output set. The first is answered precisely. The second is complicated and predicate-dependent, but is addressed in practical terms by adaptivity, which also improves the scheme's efficiency.