Editing Parallel curve (section)

{{Short description|Generalization of the concept of parallel lines}}
{{Distinguish|parallel transport}}
[[File:Offset-curves-of-sinus-curve.svg|300px|thumb|Parallel curves of the graph of <math>y=1.5 \sin(x)</math> for distances <math>d = 0.25, \dots, 1.5 </math>]]
[[File:Offset-definition-poss.svg|300px|thumb|Two definitions of a parallel curve: 1) envelope of a family of congruent circles, 2) by a fixed normal distance]]
[[File:Offset-of-a-circle.svg|thumb|The parallel curves of a circle (red) are circles, too]]

A '''parallel''' of a [[curve]] is the [[envelope (mathematics)|envelope]] of a family of [[Congruence (geometry)|congruent]] [[circle]]s centered on the curve. 
It generalises the concept of ''[[parallel (geometry)|parallel (straight) lines]]''. It can also be defined as a curve whose points are at a constant ''[[normal distance]]'' from a given curve.<ref name="Willson">{{cite book
|title=Theoretical and Practical Graphics
|first1=Frederick Newton
|last1=Willson
|publisher=Macmillan
|year=1898
|isbn=978-1-113-74312-1
|page=[https://archive.org/details/graphicscourse00willrich/page/66 66]
|url=https://archive.org/details/graphicscourse00willrich}}
</ref> 
These two definitions are not entirely equivalent as the latter assumes [[smoothness]], whereas the former does not.<ref name="DevadossO'Rourke2011">{{cite book|first1=Satyan L.|last1= Devadoss|author1-link= Satyan Devadoss |first2=Joseph |last2=O'Rourke|author2-link=Joseph O'Rourke (professor)|title=Discrete and Computational Geometry|url=https://books.google.com/books?id=InJL6iAaIQQC&pg=PA128|year=2011|publisher=Princeton University Press|isbn=978-1-4008-3898-1|pages=128–129}}</ref>

In [[computer-aided design]] the preferred term for a parallel curve is '''offset curve'''.<ref name="DevadossO'Rourke2011"/><ref name="SendraWinkler2007"/><ref name="Agoston2005m">{{cite book|first=Max K.|last= Agoston|title=Computer Graphics and Geometric Modelling: Mathematics|url=https://books.google.com/books?id=LPsAM-xuGG8C&pg=PA586|year=2005|publisher=Springer Science & Business Media|isbn=978-1-85233-817-6|page=586}}</ref> (In other geometric contexts, [[offset (disambiguation)|the term offset]] can also refer to [[Translation (geometry)|translation]].<ref name="Vince2006">{{cite book|first=John|last=Vince|title=Geometry for Computer Graphics: Formulae, Examples and Proofs|url=https://books.google.com/books?id=VkzklwKLv7UC&pg=PA293|year=2006|publisher=Springer Science & Business Media|isbn=978-1-84628-116-7|page=293}}</ref>) Offset curves are important, for example, in [[numerically controlled]] [[machining]], where they describe, for example, the shape of the cut made by a round cutting tool of a two-axis machine. The shape of the cut is offset from the trajectory of the cutter by a constant distance in the direction normal to the cutter trajectory at every point.<ref name="Marsh2006">{{cite book|first=Duncan|last=Marsh|title=Applied Geometry for Computer Graphics and CAD|url=https://books.google.com/books?id=5wHxT5W424QC&pg=PA107|year=2006|publisher=Springer Science & Business Media|isbn=978-1-84628-109-9|page=107|edition=2nd}}</ref>

In the area of 2D [[computer graphics]] known as [[vector graphics]], the (approximate) computation of parallel curves is involved in one of the fundamental drawing operations, called stroking, which is typically applied to [[polyline]]s or [[polybezier]]s (themselves called paths) in that field.<ref name="Kilgard">{{cite web |author=Mark Kilgard | title=CS 354 Vector Graphics & Path Rendering | website=www.slideshare.net | date=2012-04-10 | url=https://www.slideshare.net/Mark_Kilgard/22pathrender |page=28}}</ref>

Except in the case of a line or [[circle]], the parallel curves have a more complicated mathematical structure than the progenitor curve.<ref name="Willson"/> For example, even if the progenitor curve is [[Smooth function|smooth]], its offsets may not be so; this property is illustrated in the top figure, using a [[sine curve]] as progenitor curve.<ref name="DevadossO'Rourke2011"/> In general, even if a curve is [[rational curve|rational]], its offsets may not be so. For example, the offsets of a parabola are rational curves, but the offsets of an [[ellipse]] or of a [[hyperbola]] are not rational, even though these progenitor curves themselves are rational.<ref name="SendraWinkler2007">{{cite book|first1=J. Rafael |last1=Sendra|first2=Franz|last2= Winkler|first3=Sonia |last3=Pérez Díaz|title=Rational Algebraic Curves: A Computer Algebra Approach|url=https://books.google.com/books?id=puWxs7KG2D0C&pg=PA10|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-73724-7|page=10}}</ref>

The notion also generalizes to 3D [[surface (mathematics)|surface]]s, where it is called an '''offset surface''' or '''parallel surface'''.<ref name="Agoston2005"/> Increasing a [[solid (geometry)|solid]] volume by a (constant) distance offset is sometimes called ''dilation''.<ref name="jarek">http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf, p. 3</ref> The opposite operation is sometimes called ''shelling''.<ref name="Agoston2005">{{cite book|first=Max K.|last= Agoston|title=Computer Graphics and Geometric Modelling|url=https://books.google.com/books?id=fGX8yC-4vXUC&pg=PA645|year=2005|publisher=Springer Science & Business Media|isbn=978-1-85233-818-3|pages=638–645}}</ref> Offset surfaces are important in [[numerically controlled]] [[machining]], where they describe the shape of the cut made by a ball nose end mill of a three-axis machine.<ref name="Faux1979">{{cite book|first1=I. D.|last1=Faux|first2=Michael J.|last2=Pratt|title=Computational Geometry for Design and Manufacture|year=1979|publisher=Halsted Press|isbn=978-0-47026-473-7|oclc=4859052}}</ref> Other shapes of cutting bits can be modelled mathematically by general offset surfaces.<ref name="Brechner1990">{{cite thesis |last=Brechner|first=Eric|date=1990|title=Envelopes and tool paths for three-axis end milling|type=PhD|publisher=Rensselaer Polytechnic Institute}}</ref>