From Concentric Rings to Growth-Topology Fields: A Reflective Note on Mathematical Representation in Wood Science

This author’s research retrospective accompanies the submitted manuscript A closed-form Fourier framework for annual-ring geometry and local radial–tangential orientation from tree-disc contours. It is not part of the formal manuscript or its supplementary material and has not undergone peer review. The note records the conceptual route by which an earlier problem in wood mechanics led to the Fourier annual-ring framework. The immediate stimulus arose during the study of wood cupping, where the classical assumption of perfectly concentric annual rings became increasingly restrictive for representing eccentric and nonpolar radial–tangential material-axis fields. Controlled Fourier-type perturbations were first introduced there as synthetic descriptions of nonpolar ring geometry. This subsequently led to a broader question: if Fourier modes can represent departures from ideal rings, can observed annual-ring boundaries themselves be represented as a continuous harmonic family? The retrospective traces the resulting progression from moving concentric circles, ellipses, and mechanical geometric analogies to Fourier representation, continuous ring-coordinate fields, and local radial–tangential orientation fields. It also discusses the rediscovery of related ideas in forest mensuration, remote sensing, geometric modelling, and level-set growth theory; the role of independent cardinal-direction validation; the distinction between fixed-order Fourier under-resolution and genuine failure of a single-valued polar representation; and the interpretation of localized cusp residuals. The broader research path can be summarized as cupping mechanics → nonpolar material-axis field → Fourier ring representation → continuous growth-topology field. Its primary purpose is archival: to preserve the intellectual provenance, discarded representations, conceptual transitions, and methodological lessons that are largely invisible in the finished scientific paper. The earlier mechanics study from which the Fourier idea emerged is archived as: Kang, W. and Kang, C.W. (2026), Cross-Sectional Distortion in Polar and Nonpolar Orthotropic Lumber: A Unified Analytical and Numerical Approach. Zenodo. DOI: 10.5281/zenodo.23004879.