7. On the Problem of Determining the Distance between Parametric Curves

7.1. Introduction When comparing approximations to a given curve, it is necessary to have some comparison metric. In the functional case, a natural metric is the max-norm, which is the largest difference in y-values at corresponding points on the two curves: dist (ƒ,g)= ‖ƒ−g‖∞ ≈ max 0≤k≤n |ƒ ( xk )−g ( xk ) | , as illustrated in Fig. 7.1. This can be approximated arbitrarily well by taking a sufficiently large sample size n.