Survey on the theory and applications of μ-bases for rational curves and surfaces

Abstract

μ-Bases are new representations for rational curves and surfaces which serve as a bridge between their parametric forms and implicit forms. Geometrically, μ-bases are represented by moving lines or moving planes, while their algebraic counterparts are special syzygies of the parametric equations of rational curves or surfaces. μ-bases have been proven to be significant in solving many important problems in geometric modeling, such as fast implicitization, singularity computation, reparametrization as well as providing easy inversion formulas for points. We review the state-of-the-art results in μ-bases theory and applications for rational curves and surfaces, and raise unsolved problems for future research.