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 C1$$ {C}^1 $$ continuous interpolations within three‐dimensional spaces. However, real‐world applications often demand higher levels of smoothness within M$$ M $$‐dimensional spaces. This paper introduces LocalPoly interpolation, a novel generalization of tricubic interpolation that extends to Cn$$ {C}^n $$ continuity and M$$ M $$ 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 Cn$$ {C}^n $$ 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 C2$$ {C}^2 $$ information. The paper explores the use of LocalPoly interpolation applied to a squared distance field in ℝ3$$ {\mathbb{R}}^3 $$, 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. [ABSTRACT FROM AUTHOR]