A basis for the implicit representation of planar rational cubic B\'ezier curves

TL;DR

This paper introduces a new basis for deriving the implicit equation of planar rational cubic Bézier curves, simplifying computations and providing geometric insights, including the position of double points and singularity conditions.

Contribution

A novel basis for implicit representation of rational cubic Bézier curves is proposed, enabling explicit formulas and geometric analysis based on control points.

Findings

Explicit formula for implicit curve coefficients.

Barycentric formula for double point location.

Conditions for unwanted singularities.

Abstract

We present an approach to finding the implicit equation of a planar rational parametric cubic curve, by defining a new basis for the representation. The basis, which contains only four cubic bivariate polynomials, is defined in terms of the B\'ezier control points of the curve. An explicit formula for the coefficients of the implicit curve is given. Moreover, these coefficients lead to simple expressions which describe aspects of the geometric behaviour of the curve. In particular, we present an explicit barycentric formula for the position of the double point, in terms of the B\'ezier control points of the curve. We also give conditions for when an unwanted singularity occurs in the region of interest. Special cases in which the method fails, such as when three of the control points are collinear, or when two points coincide, will be discussed separately.