LocalPoly interpolation: Generalizing tricubic for Cn continuity in M-dimensional spaces

Tricubic interpolation, originally introduced by Lekien and Marsden (Int J Numer Methods Eng. 2005; 63(3): 455–471), has been a cornerstone in the field of interpolation, providing $C^1$ continuous interpolations within three-dimensional spaces. However, real-world applications often demand higher levels of smoothness within $M$-dimensional spaces. This paper introduces LocalPoly interpolation, a novel generalization of tricubic interpolation that extends to $C^n$ continuity and $M$ dimensions. A key property is the use of solely local data for interpolation, allowing for on-demand computation of interpolation polynomials, which is particularly advantageous in scenarios where a minor subset of the space is of interest. We rigorously prove the $C^n$ continuity achieved by the LocalPoly interpolation method; the proof features a numerically exact method for computing polynomial coefficients. The enhanced continuity is of great relevance in optimization algorithms, where efficient convergence often relies on the availability of $C^2$ information. The paper explores the use of LocalPoly interpolation applied to a squared distance field in ${ℝ}^3$, offering insights into computational efficiency and practical implications. It also discusses future research directions to address the method's limitations in terms of dimensionality, making it a valuable addition to the toolbox of interpolation methods for various scientific and engineering applications.