Editing B-spline (section)

==Relationship to piecewise/composite Bézier==
A [[Bézier curve]] is also a polynomial curve definable using a recursion from lower-degree curves of the same class and encoded in terms of control points, but a key difference is that all terms in the recursion for a Bézier curve segment have the same domain of definition (usually <math>[0, 1]</math>), whereas the [[support (mathematics)|supports]] of the two terms in the B-spline recursion are different (the outermost subintervals are not common). This means that a Bézier curve of degree <math>n</math> given by <math>m \gg n</math> control points consists of about <math>m/n</math> mostly independent segments, whereas the B-spline with the same parameters smoothly transitions from subinterval to subinterval. To get something comparable from a Bézier curve, one would need to impose a smoothness condition on transitions between segments, resulting in some manner of Bézier spline (for which many control points would be determined by the smoothness requirement).

A [[Composite Bézier curve|piecewise/composite Bézier curve]] is a series of Bézier curves joined with at least [[Smooth function#Differentiability classes|C0 continuity]] (the last point of one curve coincides with the starting point of the next curve). Depending on the application, additional smoothness requirements (such as C1 or C2 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 |pages=96–}}</ref> C1 continuous curves have identical tangents at the breakpoint (where the two curves meet). C2 continuous curves have identical curvature at the breakpoint.<ref name="Wernecke1993">{{cite book |last=Wernecke |first=Josie |date=1993 |title=The Inventor Mentor: Programming Object-Oriented 3D Graphics with Open Inventor, Release 2 |url=http://www-evasion.imag.fr/~Francois.Faure/doc/inventorMentor/sgi_html/index.html |location=Boston, MA, USA |edition=1st |publisher=Addison-Wesley Longman Publishing Co., Inc. |chapter=8 |chapter-url=http://www-evasion.imag.fr/~Francois.Faure/doc/inventorMentor/sgi_html/ch08.html |isbn=978-0201624953 }}</ref>