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→Parallel curves of an implicit curve: it appears to be untrue that the lines and circles are the only implicit curves with closed-form parallel curves |
→Parallel curves of an implicit curve: parabola |
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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>]]
Not all [[implicit curve]]s have parallel curves with analytic representations, but this is possible in some special cases. For instance, the [[Pythagorean 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 the [[parabola]]s.{{sfn|Farouki|2008|pp=216–217, 448}} For the simpler 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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