Editing Spline (mathematics) (section)

==Examples==
Suppose the interval {{math|[''a'', ''b'']}} is {{math|[0, 3]}} and the subintervals are {{math|[0, 1], [1, 2], [2, 3]}}. Suppose the polynomial pieces are to be of degree 2, and the pieces on {{math|[0, 1]}} and {{math|[1, 2]}} must join in value and first derivative (at {{math|1=''t'' = 1}}) while the pieces on {{math|[1, 2]}} and {{math|[2, 3]}} join simply in value (at {{math|1=''t'' = 2}}). This would define a type of spline {{math|''S''(''t'')}} for which

<math display=block>\begin{align}
  S(t) &= P_0 (t) = -1+4t-t^2, && 0 \le t < 1 \\[2pt]
  S(t) &= P_1 (t) = 2t, && 1 \le t < 2 \\[2pt]
  S(t) &= P_2 (t) = 2-t+t^2, && 2 \le t \le 3
\end{align}</math>

would be a member of that type, and also

<math display=block>\begin{align}
  S(t) &= P_0 (t) = -2-2t^2, && 0 \le t < 1 \\[2pt]
  S(t) &= P_1 (t) = 1-6t+t^2, && 1 \le t < 2 \\[2pt]
  S(t) &= P_2 (t) = -1+t-2t^2, && 2 \le t \le 3
\end{align}</math>

would be a member of that type.
(Note: while the polynomial piece {{math|2''t''}} is not quadratic, the result is still called a quadratic spline. This demonstrates that the degree of a spline is the maximum degree of its polynomial parts.)
The extended knot vector for this type of spline would be {{math|(0, 1, 2, 2, 3)}}.

The simplest spline has degree 0. It is also called a [[step function]].
The next most simple spline has degree 1. It is also called a '''linear spline'''. A closed linear spline (i.e, the first knot and the last are the same) in the plane is just a [[polygon]].

A common spline is the '''natural cubic spline'''.  A cubic spline has degree 3 with continuity {{math|''C''<sup>2</sup>}}, i.e. the values and first and second derivatives are continuous.  Natural means that the second derivatives of the spline polynomials are zero at the endpoints of the interval of interpolation.

<math display=block>S''(a) \, = S''(b) = 0.</math>

Thus, the graph of the spline is a straight line outside of the interval, but still smooth.