A Survey of De Casteljau Algorithms and Regular Iterative Constructions of Bézier Curves with Control Mass Points

Abstract

Drawing a curve on a computer actually involves approximating it by a set of segments. The De Casteljau algorithm allows to construct these piecewise linear curves which approximate polynomial Bézier curves using convex combinations. However, for rational Bézier curves, the construction no longer admits regular sampling. To solve this problem, we propose a generalization of the De Casteljau algorithm that addresses this issue and is applicable to Bézier curves with mass points (a weighted point or a vector) as control points and using a homographic parameter change dividing the interval [0, 1] into two equal-length intervals [0, 1/2] and [1/2 , 1] . If the initial Bézier curve is in standard form, we obtain two curves in standard form, unless the mass endpoint of the curve is a vector. This homographic parameter change also allows transforming curves defined over an interval [α, +∞], α ∈ R, into Bézier curves, which then enables the use of the De Casteljau algorithm. Some examples are given: three-quart of circle, semicircle and a branch of a hyperbola (degree 2), cubic curve on [0; +∞] and loop of a Descartes Folium (degree 3) and a loop of a Bernouilli Lemniscate (degree 4).