Parallel curve: Difference between revisions

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== Parallel curves of an implicit curve ==

[[File:Offset-of-implicit-curve-c4.svg|250px|thumb|Parallel curves of the implicit curve (red) with equation <math>x^4+y^4-1=0</math>]]

GenerallyNot theall [[implicit curve]]s have parallel curves with analytic representationrepresentations, ofbut athis parallelis curvepossible ofin ansome special cases. For instance, the [[implicitPythagorean hodograph curve]]s are rational curves with rational parallel curves, which can be converted to implicit representations. Another class of implicit rational curves with rational parallel curves is notthe possible[[parabola]]s.{{Citation Neededsfn|Farouki|2008|datepp=August216–217, 2025448}}. Only forFor the simplesimpler cases of lines and circles the parallel curves can be described easily.

For example:

: ''Line'' <math>\; f(x,y)=x+y-1=0\; </math> → distance function: <math>\; h(x,y)=\frac{x+y-1}{\sqrt{2}}=d\; </math> (Hesse normalform)

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*[http://mathworld.wolfram.com/ParallelCurves.html Parallel curves on MathWorld]

*[http://xahlee.org/SpecialPlaneCurves_dir/Parallel_dir/parallel.html Visual Dictionary of Plane Curves] Xah Lee

* http://library.imageworks.com/pdfs/imageworks-library-offset-curve-deformation-from-Skeletal-Anima.pdf application to animation; patented as {{US patent|8400455}}

* http://www2.uah.es/fsegundo/Otros/Offset/16-SanSegundoSendraSendra-1532.pdf{{Dead link|date=August 2025 |bot=InternetArchiveBot |fix-attempted=yes }}

{{Differential transforms of plane curves}}