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Given two curves, one is a parallel curve (also known as an offset curve) of the other if the points on the first curve are equidistant to the corresponding points in the direction of the second curve's normal. Alternatively, a parallel of a curve can be defined as the envelope of congruent circles whose centers lie on the curve.
For a parametric curve in the plane defined by
 , its parallel curve
 with offset  is defined by
The most elementary example of parallel curves is given by the family of concentric circles
Except for trivial cases such as circles and lines, parallel curves may be quite different from the original curve as the offset gets larger. An example of this is given by the catenary
From the definition, the family of parallel curves is then
where  correspond to the catenary.
Eliminating the parameter  from these equations; the latter gives
 , i.e.
 . Thus we obtain the implicit representation
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![\includegraphics[scale=0.5]{parallel}](http://images.planetmath.org:8080/cache/objects/9544/l2h/img1.png "\includegraphics[scale=0.5]{parallel}")
![\includegraphics[scale=0.5]{parallelc}](http://images.planetmath.org:8080/cache/objects/9544/l2h/img14.png "\includegraphics[scale=0.5]{parallelc}")
![\includegraphics[scale=0.5]{catenary}](http://images.planetmath.org:8080/cache/objects/9544/l2h/img24.png "\includegraphics[scale=0.5]{catenary}")
