Editing Spline (mathematics) (section)

==Notes==
It might be asked what meaning more than {{mvar|n}} multiple knots in a knot vector have, since this would lead to continuities like
<math display=block>S(t) \in C^{-m}, \quad m > 0</math>
at the location of this high multiplicity. By convention, any such situation indicates a simple discontinuity between the two adjacent polynomial pieces. This means that if a knot {{mvar|t{{sub|i}}}} appears more than {{math|''n'' + 1}} times in an extended knot vector, all instances of it in excess of the {{math|(''n'' + 1)}}th can be removed without changing the character of the spline, since all multiplicities {{math|''n'' + 1}}, {{math|''n'' + 2}}, {{math|''n'' + 3}}, etc. have the same meaning. It is commonly assumed that any knot vector defining any type of spline has been culled in this fashion.

The classical spline type of degree {{mvar|n}} used in numerical analysis has continuity
<math display=block>S(t) \in \mathrm{C}^{n-1} [a,b],</math>
which means that every two adjacent polynomial pieces meet in their value and first {{math|''n'' &minus; 1}} derivatives at each knot. The mathematical spline that most closely models the [[flat spline]] is a cubic ({{math|1=''n'' = 3}}), twice continuously differentiable ({{math|''C''<sup>2</sup>}}), natural spline, which is a spline of this classical type with additional conditions imposed at endpoints {{mvar|a}} and {{mvar|b}}.

Another type of spline that is much used in graphics, for example in drawing programs such as [[Adobe Illustrator]] from [[Adobe Systems]], has pieces that are cubic but has continuity only at most
<math display=block>S(t) \in C^1 [a,b].</math>
This spline type is also used in [[PostScript]] as well as in the definition of some computer typographic fonts.

Many computer-aided design systems that are designed for high-end graphics and animation use extended knot vectors,
for example [[Autodesk Maya]].
Computer-aided design systems often use an extended concept of a spline known as a [[Nonuniform rational B-spline]] (NURBS).

If sampled data from a function or a physical object is available, [[spline interpolation]] is an approach to creating a spline that approximates that data.