Computing the μ-bases of algebraic monoid curves and surfaces. (June 2021) - Pérez-Díaz, Sonia; Shen, Li-Yong
Highlights: We propose the µ-basis formulas for the implicit monoid curves/surfaces. New µ-basis formulas for the parametric monoid curves/surfaces in general expression . Approximate µ-bases are computed for the monoid curves/surfaces as well as the error estimations. Graphical abstract: Abstract: The μ -basis is a developing algebraic tool to study the expressions of rational curves and surfaces. It can play a bridge role between the parametric forms and implicit forms and show some advantages in implicitization, inversion formulas and singularity computation. However, it is difficult and there are few works to compute the μ -basis from an implicit form. In this paper, we derive the explicit forms of μ -basis for implicit monoid curves and surfaces, including the conics and quadrics which are particular cases of these entities. Additionally, we also provide the explicit form of μ -basis for monoid curves and surfaces defined by any rational parametrization (not necessarily in standard proper form). Our technique is simply based on the linear coordinate transformation and standard forms of these curves and surfaces. As a practical application in numerical situation, if an exact multiple point can not be computed, we can consider the problem of computing "approximate μ -basis" as well as the error estimation.