Average intersection and pivoting densities

For the purposes of quantifying efficiency of more-than-one codimension piecewise-linear continuation algorithms, higher dimensional analogues of the average directional density are defined and studied. Two such are explicitly considered—the average intersection density, which measures the local density of intersections of an implicitly defined piecewise-linear manifold with complementing faces of a decomposition of Euclidean space, and average pivoting density, which considers the number of pivots such an intersection entails. General results are established which relate the densities to geometric properties of the decomposition. Explicit, closed formulae are presented for the most interesting case of the Freudenthal–Kuhn triangulation and asymptotic expressions are developed.