Looking for better ways to approximate 3D data with algebraic surfaces? This article characterizes the solution space of low-degree, implicitly defined algebraic surfaces that interpolate and/or least-squares approximate scattered point and curve data in three-dimensional space. The higher-order interpolation and least-squares approximation problem reduces to a quadratic minimization problem with elegant solutions. The authors implemented their algebraic surface-fitting algorithms within the SHASTRA geometric environment and provide several examples to illustrate their application to algebraic surface design. This research provides valuable tools for geometric modeling, computer-aided design, and data visualization.
Published in ACM Transactions on Graphics, this paper aligns with the journal's focus on geometric modeling and surface representation in computer graphics. It presents algorithms for fitting algebraic surfaces to 3D data, contributing to the development of tools used in computer-aided design and other applications. The research showcases the use of algebraic techniques for solving geometric problems.