Editing Spline interpolation (section)

==Introduction==
[[Image:Cubic spline.svg|thumb|right|Interpolation with cubic splines between eight points. Hand-drawn technical drawings for shipbuilding are a historical example of spline interpolation; drawings were constructed using flexible rulers that were bent to follow pre-defined points.]]
Originally, ''[[Flat spline|spline]]'' was a term for [[wikt:elastic|elastic]] [[ruler]]s that were bent to pass through a number of predefined points, or ''knots''. These were used to make [[technical drawing]]s for [[shipbuilding]] and construction by hand, as illustrated in the figure.

We wish to model similar kinds of curves using a set of mathematical equations. Assume we have a sequence of <math>n + 1</math> knots, <math>(x_0, y_0)</math> through <math>(x_n, y_n)</math>. There will be a cubic polynomial <math>q_i(x)=y</math> between each successive pair of knots <math>(x_{i-1}, y_{i-1})</math> and <math>(x_i, y_i)</math> connecting to both of them, where <math>i = 1, 2, \dots, n</math>. So there will be <math>n</math> polynomials, with the first polynomial starting at <math>(x_0, y_0)</math>, and the last polynomial ending at <math>(x_n, y_n)</math>.

The [[curvature]] of any curve <math>y = y(x)</math> is defined as

:<math>\kappa = \frac{y''}{(1 + y'^2)^{3/2}},</math>

where <math>y'</math> and <math>y''</math> are the first and second derivatives of <math>y(x)</math> with respect to <math>x</math>.
To make the spline take a shape that minimizes the bending (under the constraint of passing through all knots), we will define both <math>y'</math> and <math>y''</math> to be continuous everywhere, including at the knots. Each successive polynomial must have equal values (which are equal to the y-value of the corresponding datapoint), derivatives, and second derivatives at their joining knots, which is to say that

:<math>\begin{cases}
 q_i(x_i) = q_{i+1}(x_i) = y_i \\
 q'_i(x_i) = q'_{i+1}(x_i) \\
 q''_i(x_i) = q''_{i+1}(x_i)
\end{cases}
\qquad
1 \le i \le n - 1.</math>

This can only be achieved if polynomials of degree 3 (cubic polynomials) or higher are used. The classical approach is to use polynomials of exactly degree 3 — [[cubic spline]]s.

In addition to the three conditions above, a ''natural cubic spline'' has the condition that <math>q''_1(x_0) = q''_n(x_n) = 0</math>.

In addition to the three main conditions above, a ''clamped cubic spline'' has the conditions that <math>q'_1(x_0) = f'(x_0)</math> and <math>q'_n(x_n) = f'(x_n)</math> where <math>f'(x)</math> is the derivative of the interpolated function.

In addition to the three main conditions above, a ''not-a-knot spline'' has the conditions that <math>q'''_1(x_1) = q'''_2(x_1)</math> and <math>q'''_{n-1}(x_{n-1}) = q'''_{n}(x_{n-1})</math>.<ref>{{Cite book |last=Burden |first=Richard |title=Numerical Analysis |last2=Faires |first2=Douglas |publisher=Cengage Learning |year=2015 |isbn=9781305253667 |edition=10th |pages=142–157}}</ref>