The geometrical construction of pointwise distributions on curves

摘要

A method is developed to generate desirable pointwise distributions along curves. This is accomplished with a simple geometrical construction which provides a global parameter for curvature clustering together with other parameters for arbitrary local clustering specifications. The level of available precision is considerable in that exact numbers of points can be assigned to both local clusters and to curvature simultaneously with specified spacing from the endpoints. The basic construction simply involves the generation of an auxiliary curve along outward normal directions from the given one. The distribution results when uniform arc-length increments are taken along the auxiliary curve and are projected back along the normals to our given curve. This construction can be applied either directly or in the form of equivalent weight functions. Moreover, it is valid regardless of whether the curve lies in Euclidian space or in surfaces and regardless of the dimensionality of the space in which the curve lies.