Recurrence for Pandimensional Space-Filling Functions

Abstract: A space-filling function is a bijection from the unit line segment to the unit square, cube, or hypercube. The function from the unit line segment is continuous. The inverse function, while well-defined, is not continuous. Space-filling curves, the finite approximations to space-filling functions, have found application in global optimization, database indexing, and dimension reduction among others. For these applications the desired transforms are mapping a scalar to multidimensional coordinates and mapping multidimensional coordinates to a scalar. Presented are recurrences which produce space-filling functions and curves of any rank $d\ge2$ based on serpentine Hamiltonian paths on $({\bf Z}\bmod s)^d$ where $s\ge2$. The recurrences for inverse space-filling functions are also presented. Both Peano and Hilbert curves and functions and their generalizations to higher dimensions are produced by these recurrences. The computations of these space-filling functions and their inverse functions are absolutely convergent geometric series. The space-filling functions are constructed as limits of integer recurrences and equivalently as non-terminating real recurrences. Scaling relations are given which enable the space-filling functions and curves and their inverses to extend beyond the unit area or volume and even to all of $d$-space. This unification of pandimensional space-filling curves facilitates quantitative comparison of curves generated from different Hamiltonian paths. The isotropy and performance in dimension reduction of a variety of space-filling curves are analyzed. For dimension reduction it is found that Hilbert curves perform somewhat better than Peano curves and their isotropic variants.