Recurrence for Pandimensional Space-Filling Functions

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 welldefined, 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 ≥ 2 based on serpentine Hamiltonian paths on (Z mod s) where s ≥ 2. 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.