Using Scaled Embedded Distances to Generate Metrics for R

We present a new class of metrics on R, which we call the scaled embedding-generated (SEG) metric. These metrics are defined with the help of a smooth one-to-one embedding of R into R and an additional scalar-valued function which is used to re-scale the distances. We describe a possible construction of an SEG metric which is based on the GaußNewton algorithm. More precisely, we show how to generate a spline embedding, which approximates a given distance graph on a finite set of points in R, in the sense of least squares. Distances are required to satisfy the generalized polygon inequality. The framework is used to define a new type of Voronoi diagram in R, which is possibly anisotropic as it allows for different distance functions for the sites. We explain a simple method to compute such Voronoi diagrams. Several examples of diagrams resulting from different SEG metrics are presented.