Thin Plate Spline Surface Reconstruction via the Method of Matched Sections

TL;DR

This paper extends the Method of Matched Sections (MMS) for surface modeling, enabling high-fidelity, smooth surface reconstruction from complex boundary conditions and sparse data, bridging computational mechanics and computer graphics.

Contribution

It demonstrates the novel application of MMS to surface reconstruction, ensuring continuity of derivatives and producing energetically optimal, aesthetically pleasing surfaces.

Findings

Successfully reconstructs complex surfaces with sparse data

Ensures continuity of curvature and shear derivatives

Produces energetically optimal, fair surfaces

Abstract

This paper further develops the Method of Matched Sections (MMS), a robust numerical framework for the solution of boundary value problems governed by partial differential equations. It demonstrates its unique applicability to the challenges of surface modeling, which lie at the intersection of computational mechanics and computer graphics. This work shows how the MMS successfully bridges this gap. By decomposing the domain into an assembly of 1D directional components matched along their entire boundaries, the method inherently enforces the continuity of all variational parameters, including second-order (curvature) and third-order (shear) derivatives. We demonstrate the method's advanced capabilities in high-fidelity surface reconstruction and blending, showing that it consistently generates energetically optimal, fair surfaces even from complex boundary conditions or sparse internal data points. By advancing the application of the MMS, this research establishes it as a powerful, physics-informed geometric tool that satisfies the dual demands of rigorous numerical analysis and aesthetic computer-aided design.