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'''Example:'''<br />
The diagram shows parallel curves of the implicit curve with equation <math>\; f(x,y)=x^4+y^4-1=0\; .</math> <br />
''Remark:''
The curves <math>\; f(x,y)=x^4+y^4-1=d\; </math> are not parallel curves, because <math>\; | \operatorname{grad} f (x,y)|=1 \;</math> is not true in the area of interest.
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</ref>
There are numerous approximation algorithms for this problem. For a 1997 survey, see Elber, Lee and Kim's "Comparing Offset Curve Approximation Methods".<ref>http://www.computer.org/csdl/mags/cg/1997/03/mcg1997030062.pdf {{Bare URL PDF|date=March 2022}}</ref>
== Parallel (offset) surfaces ==
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The problem generalizes fairly obviously to higher dimensions e.g. to offset surfaces, and slightly less trivially to [[pipe surface]]s.<ref name="PottmannWallner2001">{{cite book|first1=Helmut|last1=Pottmann|first2=Johannes|last2=Wallner|title=Computational Line Geometry|url=https://books.google.com/books?id=6ZrqcYKgtE0C&pg=PA303|year=2001|publisher=Springer Science & Business Media|isbn=978-3-540-42058-3|pages=303–304}}</ref> Note that the terminology for the higher-dimensional versions varies even more widely than in the planar case, e.g. other authors speak of parallel fibers, ribbons, and tubes.<ref name="Chirikjian2009">{{cite book|first=Gregory S.|last=Chirikjian|title=Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods|year=2009|publisher=Springer Science & Business Media|isbn=978-0-8176-4803-9|pages=171–175}}</ref> For curves embedded in 3D surfaces the offset may be taken along a [[geodesic]].<ref name="Sarfraz2003">{{cite book|editor-first=Muhammad|editor-last=Sarfraz|title=Advances in geometric modeling|url=https://books.google.com/books?id=kfZQAAAAMAAJ&pg=PA72|year=2003|publisher=Wiley|isbn=978-0-470-85937-7|page=72}}</ref>
Another way to generalize it is (even in 2D) to consider a variable distance, e.g. parametrized by another curve.<ref name="barn"/> One can for example stroke (envelope) with an ellipse instead of circle<ref name="barn"/> as it is possible for example in [[METAFONT]].<ref>https://www.tug.org/TUGboat/tb16-3/tb48kinc.pdf {{Bare URL PDF|date=March 2022}}</ref>
[[File:Envelope of ellipses.png|thumb|An envelope of ellipses forming two general offset curves above and below a given curve]]
More recently [[Adobe Illustrator]] has added somewhat similar facility in version [[CS5]], although the control points for the variable width are visually specified.<ref>http://design.tutsplus.com/tutorials/illustrator-cs5-variable-width-stroke-tool-perfect-for-making-tribal-designs--vector-4346 application of the generalized version in Adobe Illustrator CS5 (also [http://tv.adobe.com/watch/learn-illustrator-cs5/using-variablewidth-strokes/ video])</ref> In contexts where it's important to distinguish between constant and variable distance offsetting the acronyms CDO and VDO are sometimes used.<ref name="jarek"/>
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