Using the Fourier Polynomial with Fractional Terms for Implicit Surface Modeling

Abstract

Implicit surfaces are widely used in computer modeling due to their ability to model complex objects with relatively little data. Also, in contrast with explicit ways of modelling, such as with triangle meshes and boundary representation of parametric surfaces, the implicit surfaces are guaranteed to keep continuity and smoothness under the operations of deformation or replication.

The explicit modeling, however, such as triangle mesh or boundary representation of parametric surfaces, is the de facto current standard of computer modeling. The problem of conversion from the explicit representation to an implicit one is therefore a cutting-edge engineering problem. New approaches and specific methods are published every year, including those that are based on machine learning or artificial intelligence in general.

In this work, we explore the limits of a triangle mesh to implicit surface conversion based on Fourier polynomial approximation, although the polynomial we propose contains fractional terms. The prospects of using a polynomial with solely integer terms are limited by the Fourier polynomial`s tendency to overfit. We show that adding fractional terms to the integer ones in a single function allows for reaching the modeling quality that could be satisfactory for some practical scenarios, while keeping the total amount of polynomial coefficients relatively small too.

In theory, using the Fourier polynomial with fractional terms could be used not only for smooth surface modeling but for smooth connection of different triply periodic surfaces along a selected plane due to the planned loss of periodicity along the normal axis. This, however, requires studying the question of specific per-axis polynomial configuration, which extends beyond the scope of this particular paper.