WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 23, 2024
A Survey of De Casteljau Algorithms and Regular Iterative Constructions of Bézier Curves with Control Mass Points
Authors: , ,
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 $$[0, \frac{1}{2}]$$ and $$[\frac{1}{2},1]$$ 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 $$[α,+\infty ]$$, $$α \in \mathbb{R}$$, 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 $$[0;+\infty]$$ and loop of a Descartes Folium (degree 3) and a loop of
a Bernouilli Lemniscate (degree 4).
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Keywords: De Casteljau algorithms, Bézier curves, mass points, homographic parameter change,
regular sampling iterative construction
Pages: 216-236
DOI: 10.37394/23206.2024.23.25