Distances between oriented curves in geometric modeling (Q1371975)

The authors formulate conditions for geometrically interesting metrics in the space of piecewise continuous curves in a given Euclidean space. Apart from the usual conditions in this case they replace the triangle inequality by some non-Archimedean condition (a weak and a strong case) applied to concatenation of curves (that is adapted to the needs of CAD). They show that in general these conditions are inconsistent with the triangle inequality and give examples in which the Hausdorff metric is inadequate for the needs of CAD. Their candidate for a geometric distance of curves is the Fréchet metric known from the calculus of variations and analogous Fréchet-\(L_p\) metrics. A number of useful properties and alternative representations of these metrics are derived.