Spline interpolation

Introduction

Elastic rulers that were bent to pass through a number of predefined points (the "knots") were used for making technical drawings for shipbuilding and construction by hand, as illustrated by figure 1.

The approach to mathematically model the shape of such elastic rulers fixed by n+1 "knots" $(x_i,y_i)\quad i=0,1,\cdots ,n$ is to interpolate between all the pairs of "knots" $(x_{i-1}\ ,\ y_{i-1})$ and $(x_i\ ,\ y_i)$ with polynomials $y=q_i(x) \quad i=1,2,\cdots ,n$

The curvature of a curve

$y=f(x)$

$\kappa= \frac{y''}{(1%2By'^2)^{3/2}}$

As the elastic ruler will take a shape that minimizes the bending under the constraint of passing through all "knots" both $y'$ and $y''$ will be continuous everywhere, also at the "knots". To achieve this one must have that

$q'_i(x_i) = q'_{i%2B1}(x_i)$

and that

$q''_i(x_i) = q''_{i%2B1}(x_i)$

for all i , $1 \le i \le n-1$ . This can only be achieved if polynomials of degree 3 or higher are used. The classical approach is to use polynomials of degree 3, this is the case of "Cubic splines".

Algorithm to find the interpolating cubic spline

A third order polynomial $q(x)$ for which

$q(x_1)=y_1$

$q(x_2)=y_2$

$q^'(x_1)=k_1$

$q^'(x_2)=k_2$

can be written in the symmetrical form

$q\ =\ (1-t)\ y_1 %2B\ t\ y_2\ %2B\ t\ (1-t)\ (a\ (1-t) %2B b\ t)$

(1)

where

$t=\frac{x-x_1}{x_2-x_1}$

(2)

and

$a= k_1 (x_2 - x_1)-(y_2 - y_1)$

(3)

$b=-k_2 (x_2 - x_1)%2B(y_2 - y_1)$

(4)

As $q^'= \frac{d q}{d x} = \frac{d q}{d t} \ \frac{d t}{d x} = \frac{d q}{d t} \ \frac{1}{x_2-x_1}$ one gets that

$q^'\ =\frac {y_2-y_1}{x_2-x_1} %2B(1-2t)\ \frac {a\ (1-t) %2B b\ t}{x_2-x_1}\ %2B\ \ t\ (1-t)\ \frac {b-a}{x_2-x_1}$

(5)

$q^{''}=2\frac {b-2a%2B(a-b)3t}{{(x_2-x_1)}^2}$

(6)

Setting $x=x_1$ and $x=x_2$ in (5) and (6) one gets from (2) that indeed $q^'(x_1)= k_1$ , $q^'(x_2) = k_2$ and that

$q^{''}(x_1)=2\frac {b-2a}{{(x_2-x_1)}^2}$

(7)

$q^{''}(x_2)=2\frac {a-2b}{{(x_2-x_1)}^2}$

(8)

If now

$(x_i,y_i)\quad i=0,1,\cdots ,n$

are n+1 points and

$q_i\ =\ (1-t)\ y_{i-1} %2B\ t\ y_i\ %2B\ t\ (1-t)\ (a_i\ (1-t) %2B b_i\ t)\quad i=1,\cdots ,n$

(9)

where

$t=\frac{x-x_{i-1}}{x_i-x_{i-1}}$

are n third degree polynomials interpolating $y$ in the interval $x_{i-1} \le x<x_i \le$ , for $i=1,\cdots ,n$ such that

$q^'_i(x_i)=q^'_{i%2B1}(x_i)$

for $i=1,\cdots ,n-1$

then the n polynomials together define a derivable function in the interval $x_0 \le x \le x_n$ and

$a_i=k_{i-1}(x_i-x_{i-1})-(y_i - y_{i-1})$

(10)

$b_i=-k_i(x_i-x_{i-1})%2B(y_i - y_{i-1})$

(11)

for $i=1,\cdots ,n$ where

$k_0=q_1^'(x_0)$

(12)

$k_i=q_i^'(x_i)=q_{i%2B1}^'(x_i) \quad i=1,\cdots ,n-1$

(13)

$k_n=q_n^'(x_n)$

(14)

If the sequence $k_0,k_1, \cdots ,k_n$ is such that in addition

$q^{''}_i(x_i)=q^{''}_{i%2B1}(x_i)$

for $i=1,\cdots ,n-1$

the resulting function will even have a continuous second derivative.

From (7), (8), (10) and (11) follows that this is the case if and only if

$\frac {k_{i-1}}{x_i-x_{i-1}} %2B \left(\frac {1}{x_i-x_{i-1}}%2B \frac {1}{{x_{i%2B1}-x_i}}\right)\ 2k_i%2B \frac {k_{i%2B1}}{{x_{i%2B1}-x_i}} = 3\ \left(\frac {y_i - y_{i-1}}{{(x_i-x_{i-1})}^2}%2B\frac {y_{i%2B1} - y_i}{{(x_{i%2B1}-x_i)}^2}\right)$

(15)

for $i=1,\cdots ,n-1$

The relations (15) are n-1 linear equations for the n+1 values $k_0,k_1, \cdots ,k_n$ .

For the elastic rulers being the model for the spline interpolation one has that to the left of the left-most "knot" and to the right of the right-most "knot" the ruler can move freely and will therefore take the form of a straight line with $q''= 0$ . As $q''$ should be a continuous function of $x$ one gets that for "Natural Splines" one in addition to the n-1 linear equations (15) should have that

$q^{''}_i(x_0)\ =2\ \frac {3(y_1 - y_0)-(k_1%2B2k_0)(x_1-x_0)}{{(x_1-x_0)}^2}=0$

$q^{''}_n(x_n)\ =-2\ \frac {3(y_n - y_{n-1})-(2k_n%2Bk_{n-1})(x_n-x_{n-1})}{{(x_n-x_{n-1})}^2}=0$

i.e. that

$\frac{2}{x_1-x_0} k_0\ %2B\frac{1}{x_1-x_0}k_1 = 3\ \frac{y_1-y_0}{(x_1-x_0)^2}$

(16)

$\frac{1}{x_n-x_{n-1}}k_{n-1}\ %2B\frac{2}{x_n-x_{n-1}}k_n = 3\ \frac{y_n-y_{n-1}}{(x_n-x_{n-1})^2}$

(17)

(15) together with (16) and (17) constitute n+1 linear equations that uniquely define the n+1 parameters $k_0,k_1, \cdots ,k_n$

Example

In case of three points the values for $k_0,k_1,k_2$ are found by solving the linear equation system

$\begin{bmatrix} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & a_{23} \\ 0 & a_{32} & a_{33} \\ \end{bmatrix} \begin{bmatrix} k_0 \\ k_1 \\ k_2 \\ \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\ \end{bmatrix}$