Thin Plate Spline Surface Reconstruction via the Method of Matched Sections

TL;DR

This research establishes the Method of Matched Sections as a powerful, physics-informed geometric tool that satisfies the dual demands of rigorous numerical analysis and aesthetic computer-aided design.

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, specifically for surface modeling. While originating in mechanics, the method addresses critical challenges in isogeometric analysis and computer graphics by bridging the gap between physical accuracy and geometric continuity. 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 points. By advancing the application of 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.

Figures (19)

• Figure 1: General scheme of a finite element decomposition
• Figure 2: Square domain with singular corner constraints
• Figure 3: Corner element topology
• Figure 4: $15 \times 15$ mesh with boundary scaling
• Figure 5: Cosine-like biharmonic surface patch