Parallel curve: Difference between revisions

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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'''.<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|url=https://www.worldcat.org/title/computational-geometry-for-design-and-manufacture/oclc/4859052|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 ==

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* {{cite book|first=Ian R. |last=Porteous|title=Geometric Differentiation: For the Intelligence of Curves and Surfaces|year=2001|publisher=Cambridge University Press|isbn=978-0-521-00264-6|pages=1–25|edition=2nd}}

* {{cite book|first1=Nicholas M. |last1=Patrikalakis|first2=Takashi |last2=Maekawa|title=Shape Interrogation for Computer Aided Design and Manufacturing|year=2010|origyear=2002|publisher=Springer Science & Business Media|isbn=978-3-642-04074-0|at=Chapter 11. Offset Curves and Surfaces}} [http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node210.html Free online version].

* {{cite conference |first=François |last=Anton |first2=Ioannis Z. |last2=Emiris |first3=Bernard |last3=Mourrain |first4=Monique |last4=Teillaud | author4-link=Monique Teillaud |article=The O set to an Algebraic Curve and an Application to Conics |title=International Conference on Computational Science and its Applications |date=May 2005 |___location=Singapore |publisher=Springer Verlag |pp=683-696683–696}}

* {{cite book |first=Rida T. |last=Farouki |title=Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable |year=2008 |publisher=Springer Science & Business Media |isbn=978-3-540-73397-3 |pp=141–178}} Pages listed are the general and introductory material.

* {{cite book |editor-first=Y.-S. |editor-last=Ma |title=Semantic Modeling and Interoperability in Product and Process Engineering: A Technology for Engineering Informatics |year=2013 |publisher=Springer Science & Business Media |isbn=978-1-4471-5073-2 |chapter=Computation of Offset Curves Using a Distance Function: Addressing a Key Challenge in Cutting Tool Path Generation |first1=C. K. |last1=Au |first2=Y.-S. |last2=Ma |pages=259–273}}