Editing Parallel curve (section)

== 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>]]
Generally the analytic representation of a parallel curve of an [[implicit curve]] is not possible. Only for the simple 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)
: ''Circle'' <math>\; f(x,y)=x^2+y^2-1=0\;</math> → distance function: <math>\; h(x,y)=\sqrt{x^2+y^2}-1=d\; .</math>

In general, presuming certain conditions, one can prove the existence of an [[oriented distance function]] <math>h(x,y)</math>. In practice one has to treat it numerically.<ref>E. Hartmann: [http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf ''Geometry and Algorithms for COMPUTER AIDED DESIGN.''] S. 81, S. 30, 41, 44.</ref> Considering parallel curves the following is true:
* The parallel curve for distance  d is the [[level set]] <math>h(x,y)=d</math> of the corresponding oriented distance function <math>h</math>.

===Properties of the distance function:<ref name="hart30" /><ref>{{cite book | last=Thorpe | first=John A. | title=Elementary Topics in Differential Geometry | publisher=Springer Science & Business Media | publication-place=New York Heidelberg | date=1994-10-27 | isbn=0-387-90357-7 | page=}}</ref>===
*<math>| \operatorname{grad}  h (\vec x)|=1 \; ,</math>
* <math> h(\vec x+d\operatorname{grad} h (\vec x)) = h(\vec x)+d \; ,</math>
*<math> \operatorname{grad}h(\vec x+d\operatorname{grad}h (\vec x))=  \operatorname{grad}h (\vec x) \; .</math>

'''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.