Content deleted Content added
Fgnievinski (talk | contribs) |
Fgnievinski (talk | contribs) |
||
Line 23:
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 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>
== Parallel curve of a parametrically given curve ==
| |||