Editing Composite Bézier curve (section)

{{Short description|Geometric shape}}
{{Use dmy dates|date=January 2021}}
[[Image:Beziergon.svg|thumb|right|200px|Béziergon – The red béziergon passes through the blue vertices, the green points are control points that determine the shape of the connecting Bézier curves]]
In [[geometric modelling]] and in [[computer graphics]], a '''composite Bézier curve''' or '''Bézier spline''' is a [[Spline (mathematics)|spline]] made out of [[Bézier curve]]s that is at least <math>C^0</math> [[continuous function|continuous]]. In other words, a composite Bézier curve is a series of Bézier curves joined end to end where the last point of one curve coincides with the starting point of the next curve. Depending on the application, additional smoothness requirements (such as <math>C^1</math> or <math>C^2</math> continuity) may be added.<ref name="ShikinPlis1995">{{cite book|author1=Eugene V. Shikin|author2=Alexander I. Plis|title=Handbook on Splines for the User|url=https://books.google.com/books?id=DL88KouJCQkC&pg=PA96|date=14 July 1995|publisher=CRC Press|isbn=978-0-8493-9404-1|page=96}}</ref>

A continuous composite Bézier is also called a '''polybézier''', by similarity to [[polyline]], but whereas in polylines the points are connected by straight lines, in a polybézier the points are connected by Bézier curves. A '''béziergon''' (also called '''bézigon''') is a closed path composed of [[Bézier curve]]s. It is similar to a [[polygon]] in that it connects a set of [[vertex (geometry)|vertices]] by lines, but whereas in polygons the vertices are connected by straight lines, in a béziergon the vertices are connected by Bézier curves.<ref>[http://msdn2.microsoft.com/en-us/library/ms534244.aspx Microsoft polybezier API]
</ref><ref>[http://libpapyrus.sourceforge.net/guide_beziergon.html Papyrus béziergon API reference]
</ref><ref>[https://books.google.com/books?id=nFAEAAAAMBAJ&dq=bezigon+curve&pg=PA85 "A better box of crayons"].
InfoWorld.
1991.</ref> Some authors even call a <math>C^0</math> composite Bézier curve a "Bézier spline";<ref>{{Cite book|url=https://books.google.com/books?id=lFwXglfyoIQC|title=A First Course in Applied Mathematics|last=Rebaza|first=Jorge|date=2012-04-24|publisher=John Wiley & Sons|isbn=9781118277157|language=en}}</ref> the latter term is however used by other authors as a synonym for the (non-composite) Bézier curve, and they add "composite" in front of "Bézier spline" to denote the composite case.<ref>{{Cite book|url=https://books.google.com/books?id=VnH0UzTycTcC|title=Mathematica ® 3.0 Standard Add-on Packages|last=(Firm)|first=Wolfram Research|date=1996-09-13|publisher=Cambridge University Press|isbn=9780521585859|language=en}}</ref>

Perhaps the most common use of composite Béziers is to describe the outline of each letter in a [[PostScript]] or [[PDF]] file. Such outlines are composed of one béziergon for [[typeface anatomy|open letters]], or multiple béziergons for closed letters. Modern [[vector graphics]] and [[computer font]] systems like [[PostScript]], [[Asymptote (vector graphics language)|Asymptote]], [[Metafont]], [[OpenType]], and [[Scalable Vector Graphics|SVG]] use composite Bézier curves composed of cubic Bézier curves (3rd order curves) for drawing curved shapes.

[[File:Sinc Function Approximation with Bezier Splines.svg|thumb|[[Sinc]] function approximated using a smooth Bézier spline, i.e., a series of smoothly-joined Bézier curves]]