Moving Planes and Singular Points of Rational Parametric Surfaces

TL;DR

This paper explores the connection between moving planes and singular points on rational parametric surfaces, introducing new definitions and relationships that enhance understanding of surface singularities and their algebraic properties.

Contribution

It introduces a new definition for the order of singular points on rational surfaces and establishes their relationship with moving planes and the μ-basis.

Findings

Derived an equivalent definition for singularity order.

Established the relationship between moving planes and singularity order.

Discussed the connection between μ-basis and singular points.

Abstract

In this paper we discuss the relationship between the moving planes of a rational parametric surface and the singular points on it. Firstly, the intersection multiplicity of several planar curves is introduced. Then we derive an equivalent definition for the order of a singular point on a rational parametric surface. Based on the new definition of singularity orders, we derive the relationship between the moving planes of a rational surface and the order of singular points. Especially, the relationship between the $\mu$-basis and the order of a singular point is also discussed.