Editing Parallel curve (section)

== Algorithms ==
{{expand section|date=August 2014}}

In general, the parallel curve of a [[Bézier curve]] is not another Bézier curve, a result proved by Tiller and Hanson in 1984.<ref>{{cite journal |last1=Tiller |first1=Wayne |last2=Hanson |first2=Eric |title=Offsets of Two-Dimensional Profiles |journal=IEEE Computer Graphics and Applications |year=1984 |volume=4 |issue=9 |pages=36–46 |doi=10.1109/mcg.1984.275995|s2cid=9046817 }}</ref> Thus, in practice, approximation techniques are used. Any desired level of accuracy is possible by repeatedly subdividing the curve, though better techniques require fewer subdivisions to attain the same level of accuracy. A 1997 survey by Elber, Lee and Kim<ref>{{cite journal |last1=Elber |first1=Gershon |last2=Lee |first2=In-Kwon |last3=Kim |first3=Myung-Soo 
|url=https://ieeexplore.ieee.org/document/586019 |doi=10.1109/38.586019 |title=Comparing offset curve approximation methods |journal=IEEE Computer Graphics and Applications |volume=17 |issue=3 |pages=62–71 |date=May–Jun 1997}}</ref> is widely cited, though better techniques have been proposed more recently. A modern technique based on [[curve fitting]], with references and comparisons to other algorithms, as well as open source JavaScript source code, was published in a blog post<ref>{{cite web |url=https://raphlinus.github.io/curves/2022/09/09/parallel-beziers.html |title=Parallel curves of cubic Béziers |last=Levien |first=Raph |date=September 9, 2022 |access-date=September 9, 2022}}</ref> in September 2022.

Another efficient algorithm for offsetting is the level approach described by
[[Ron Kimmel|Kimmel]] and Bruckstein (1993).<ref>{{cite journal | last1=Kimmel | first1=R. | last2=Bruckstein | first2=A.M. | title=Shape offsets via level sets | journal=Computer-Aided Design | publisher=Elsevier BV | volume=25 | issue=3 | year=1993 | issn=0010-4485 | doi=10.1016/0010-4485(93)90040-u | pages=154–162 | s2cid=8434463 |url=https://www.cs.technion.ac.il/~ron/PAPERS/offsets_cad1993.pdf}}</ref>