spline interpolation

Image:Cubic spline.svg

Originally, spline was a term for elastic rulers that were bent to pass through a number of predefined points, or knots. These were used to make technical drawings 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 $n\; +\; 1$ knots, $(x\_0,\; y\_0)$ through $(x\_n,\; y\_n)$. There will be a cubic polynomial $q\_i(x)=y$ between each successive pair of knots $(x\_\{i-1\},\; y\_\{i-1\})$ and $(x\_i,\; y\_i)$ connecting to both of them, where $i\; =\; 1,\; 2,\; \backslash dots,\; n$. So there will be $n$ polynomials, with the first polynomial starting at $(x\_0,\; y\_0)$, and the last polynomial ending at $(x\_n,\; y\_n)$.

The curvature of any curve $y\; =\; y(x)$ is defined as

:$\backslash kappa\; =\; \backslash frac\{y\text{'}\text{'}\}\{(1\; +\; y\text{'}^2)^\{3/2\}\},$

where $y\text{'}$ and $y\text{'}\text{'}$ are the first and second derivatives of $y(x)$ with respect to $x$.

To make the spline take a shape that minimizes the bending (under the constraint of passing through all knots), we will define both $y\text{'}$ and $y\text{'}\text{'}$ 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

:$\backslash begin\{cases\}$

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.

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 splines.

In addition to the three conditions above, a natural cubic spline has the condition that $q$_1(x_0) = q_n(x_n) = 0.

In addition to the three main conditions above, a clamped cubic spline has the conditions that $q\text{'}\_1(x\_0)\; =\; f\text{'}(x\_0)$ and $q\text{'}\_n(x\_n)\; =\; f\text{'}(x\_n)$ where $f\text{'}(x)$ is the derivative of the interpolated function.

In addition to the three main conditions above, a not-a-knot spline has the conditions that $q$_1(x_1) = q_2(x_1) and $q$_{n-1}(x_{n-1}) = q_{n}(x_{n-1}).{{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}}

We wish to find each polynomial $q\_i(x)$ given the points $(x\_0,\; y\_0)$ through $(x\_n,\; y\_n)$. To do this, we will consider just a single piece of the curve, $q(x)$, which will interpolate from $(x\_1,\; y\_1)$ to $(x\_2,\; y\_2)$. This piece will have slopes $k\_1$ and $k\_2$ at its endpoints. Or, more precisely,

:$q(x\_1)\; =\; y\_1,$

:$q(x\_2)\; =\; y\_2,$

:$q\text{'}(x\_1)\; =\; k\_1,$

:$q\text{'}(x\_2)\; =\; k\_2.$

The full equation $q(x)$ can be written in the symmetrical form

{{NumBlk|:|$q(x)\; =\; \backslash big(1\; -\; t(x)\backslash big)\backslash ,y\_1\; +\; t(x)\backslash ,y\_2\; +\; t(x)\backslash big(1\; -\; t(x)\backslash big)\backslash Big(\backslash big(1\; -\; t(x)\backslash big)\backslash ,a\; +\; t(x)\backslash ,b\backslash Big),$|{{EquationRef|1}}}}

where

{{NumBlk|:|$t(x)\; =\; \backslash frac\{x\; -\; x\_1\}\{x\_2\; -\; x\_1\},$|{{EquationRef|2}}}}

{{NumBlk|:|$a\; =\; k\_1\; (x\_2\; -\; x\_1)\; -\; (y\_2\; -\; y\_1),$|{{EquationRef|3}}}}

{{NumBlk|:|$b\; =\; -k\_2\; (x\_2\; -\; x\_1)\; +\; (y\_2\; -\; y\_1).$|{{EquationRef|4}}}}

But what are $k\_1$ and $k\_2$? To derive these critical values, we must consider that

:$q\text{'}\; =\; \backslash frac\{dq\}\{dx\}\; =\; \backslash frac\{dq\}\{dt\}\; \backslash frac\{dt\}\{dx\}\; =\; \backslash frac\{dq\}\{dt\}\; \backslash frac\{1\}\{x\_2\; -\; x\_1\}.$

It then follows that

{{NumBlk|:|$q\text{'}\; =\; \backslash frac\{y\_2\; -\; y\_1\}\{x\_2\; -\; x\_1\}\; +\; (1\; -\; 2t)\; \backslash frac\; \{a(1\; -\; t)\; +\; bt\}\{x\_2\; -\; x\_1\}\; +\; t(1\; -\; t)\; \backslash frac\{b\; -\; a\}\{x\_2\; -\; x\_1\},$|{{EquationRef|5}}}}

{{NumBlk|:|$q\text{'}\text{'}\; =\; 2\; \backslash frac\{b\; -\; 2a\; +\; (a\; -\; b)3t\}\{\{(x\_2\; -\; x\_1)\}^2\}.$|{{EquationRef|6}}}}

Setting {{math|t {{=}} 0}} and {{math|t {{=}} 1}} respectively in equations ({{EquationNote|5}}) and ({{EquationNote|6}}), one gets from ({{EquationNote|2}}) that indeed first derivatives {{math|q′(x₁) {{=}} k₁}} and {{math|q′(x₂) {{=}} k₂}}, and also second derivatives

{{NumBlk|:|$q\text{'}\text{'}(x\_1)\; =\; 2\; \backslash frac\{b\; -\; 2a\}\{\{(x\_2\; -\; x\_1)\}^2\},$|{{EquationRef|7}}}}

{{NumBlk|:|$q\text{'}\text{'}(x\_2)\; =\; 2\; \backslash frac\{a\; -\; 2b\}\{\{(x\_2\; -\; x\_1)\}^2\}.$|{{EquationRef|8}}}}

If now {{math|(x_i, y_i), i {{=}} 0, 1, ..., n}} are {{math|n + 1}} points, and

{{NumBlk|:|$q\_i\; =\; (1\; -\; t)\backslash ,y\_\{i-1\}\; +\; t\backslash ,y\_i\; +\; t(1\; -\; t)\backslash big((1\; -\; t)\backslash ,a\_i\; +\; t\backslash ,b\_i\backslash big),$|{{EquationRef|9}}}}

where i = 1, 2, ..., n, and $t\; =\; \backslash tfrac\{x\; -\; x\_\{i-1\}\}\{x\_i\; -\; x\_\{i-1\}\}$ are n third-degree polynomials interpolating {{mvar|y}} in the interval {{math|x_i−1 ≤ x ≤ x_i}} for i = 1, ..., n such that {{math|q′_i (x_i) {{=}} q′_i+1(x_i)}} for i = 1, ..., n − 1, then the n polynomials together define a differentiable function in the interval {{math|x₀ ≤ x ≤ x_n}}, and

{{NumBlk|:|$a\_i\; =\; k\_\{i-1\}(x\_i\; -\; x\_\{i-1\})\; -\; (y\_i\; -\; y\_\{i-1\}),$|{{EquationRef|10}}}}

{{NumBlk|:|$b\_i\; =\; -k\_i(x\_i\; -\; x\_\{i-1\})\; +\; (y\_i\; -\; y\_\{i-1\})$|{{EquationRef|11}}}}

for i = 1, ..., n, where

{{NumBlk|:|$k\_0\; =\; q\_1\text{'}(x\_0),$|{{EquationRef|12}}}}

{{NumBlk|:|$k\_i\; =\; q\_i\text{'}(x\_i)\; =\; q\_\{i+1\}\text{'}(x\_i),\; \backslash qquad\; i\; =\; 1,\; \backslash dots,\; n\; -\; 1,$|{{EquationRef|13}}}}

{{NumBlk|:|$k\_n\; =\; q\_n\text{'}(x\_n).$|{{EquationRef|14}}}}

If the sequence {{math|k₀, k₁, ..., k_n}} is such that, in addition, {{math|q′′_i(x_i) {{=}} q′′_i+1(x_i)}} holds for i = 1, ..., n − 1, then the resulting function will even have a continuous second derivative.

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

{{NumBlk|:|$\backslash frac\{k\_\{i-1\}\}\{x\_i\; -\; x\_\{i-1\}\}\; +\; \backslash left(\backslash frac\{1\}\{x\_i\; -\; x\_\{i-1\}\}\; +\; \backslash frac\{1\}\{\{x\_\{i+1\}\; -\; x\_i\}\}\backslash right)\; 2k\_i\; +\; \backslash frac\{k\_\{i+1\}\}\{\{x\_\{i+1\}\; -\; x\_i\}\}\; =\; 3\; \backslash left(\backslash frac\{y\_i\; -\; y\_\{i-1\}\}\{\{(x\_i\; -\; x\_\{i-1\})\}^2\}\; +\; \backslash frac\{y\_\{i+1\}\; -\; y\_i\}\{\{(x\_\{i+1\}\; -\; x\_i)\}^2\}\backslash right)$|{{EquationRef|15}}}}

for i = 1, ..., n − 1. The relations ({{EquationNote|15}}) are {{math|n − 1}} linear equations for the {{math|n + 1}} values {{math|k₀, k₁, ..., 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 {{math|q′′ {{=}} 0}}. As {{mvar|q′′}} should be a continuous function of {{mvar|x}}, "natural splines" in addition to the {{math|n − 1}} linear equations ({{EquationNote|15}}) should have

:$q\text{'}\text{'}\_1(x\_0)\; =\; 2\; \backslash frac\; \{3(y\_1\; -\; y\_0)\; -\; (k\_1\; +\; 2k\_0)(x\_1\; -\; x\_0)\}\{\{(x\_1\; -\; x\_0)\}^2\}\; =\; 0,$

:$q\text{'}\text{'}\_n(x\_n)\; =\; -2\; \backslash frac\; \{3(y\_n\; -\; y\_\{n-1\})\; -\; (2k\_n\; +\; k\_\{n-1\})(x\_n\; -\; x\_\{n-1\})\}\{\{(x\_n\; -\; x\_\{n-1\})\}^2\}\; =\; 0,$

i.e. that

{{NumBlk|:|$\backslash frac\{2\}\{x\_1\; -\; x\_0\}\; k\_0\; +\; \backslash frac\{1\}\{x\_1\; -\; x\_0\}\; k\_1\; =\; 3\; \backslash frac\{y\_1\; -\; y\_0\}\{(x\_1\; -\; x\_0)^2\},$|{{EquationRef|16}}}}

{{NumBlk|:|$\backslash frac\{1\}\{x\_n\; -\; x\_\{n-1\}\}k\_\{n-1\}\; +\; \backslash frac\{2\}\{x\_n\; -\; x\_\{n-1\}\}\; k\_n\; =\; 3\; \backslash frac\{y\_n\; -\; y\_\{n-1\}\}\{(x\_n\; -\; x\_\{n-1\})^2\}.$|{{EquationRef|17}}}}

Eventually, ({{EquationNote|15}}) together with ({{EquationNote|16}}) and ({{EquationNote|17}}) constitute {{math|n + 1}} linear equations that uniquely define the {{math|n + 1}} parameters {{math|k₀, k₁, ..., k_n}}.

There exist other end conditions, "clamped spline", which specifies the slope at the ends of the spline, and the popular "not-a-knot spline", which requires that the third derivative is also continuous at the {{math|x₁}} and {{math|x_n−1}} points.

For the "not-a-knot" spline, the additional equations will read:

:$q$_1(x_1) = q_2(x_1) \Rightarrow \frac{1}{\Delta x_1^2} k_0 + \left( \frac{1}{\Delta x_1^2} - \frac{1}{\Delta x_2^2} \right) k_1 - \frac{1}{\Delta x_2^2} k_2 = 2 \left( \frac{\Delta y_1}{\Delta x_1^3} - \frac{\Delta y_2}{\Delta x_2^3} \right),

:$q$_{n-1}(x_{n-1}) = q_n(x_{n-1}) \Rightarrow \frac{1}{\Delta x_{n-1}^2} k_{n-2} + \left( \frac{1}{\Delta x_{n-1}^2} - \frac{1}{\Delta x_n^2} \right) k_{n-1} - \frac{1}{\Delta x_n^2} k_n = 2\left( \frac{\Delta y_{n-1} }{\Delta x_{n-1}^3 }- \frac{ \Delta y_n}{ \Delta x_n^3 } \right),

where $\backslash Delta\; x\_i\; =\; x\_i\; -\; x\_\{i-1\},\backslash \; \backslash Delta\; y\_i\; =\; y\_i\; -\; y\_\{i-1\}$.

Image:Cubic splines three points.svg

In case of three points the values for $k\_0,\; k\_1,\; k\_2$ are found by solving the tridiagonal 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}

with

:$a\_\{11\}\; =\; \backslash frac\{2\}\{x\_1\; -\; x\_0\},$

:$a\_\{12\}\; =\; \backslash frac\{1\}\{x\_1\; -\; x\_0\},$

:$a\_\{21\}\; =\; \backslash frac\{1\}\{x\_1\; -\; x\_0\},$

:$a\_\{22\}\; =\; 2\; \backslash left(\backslash frac\{1\}\{x\_1\; -\; x\_0\}\; +\; \backslash frac\{1\}\{\{x\_2\; -\; x\_1\}\}\backslash right),$

:$a\_\{23\}\; =\; \backslash frac\{1\}\{\{x\_2\; -\; x\_1\}\},$

:$a\_\{32\}\; =\; \backslash frac\{1\}\{x\_2\; -\; x\_1\},$

:$a\_\{33\}\; =\; \backslash frac\{2\}\{x\_2\; -\; x\_1\},$

:$b\_1\; =\; 3\; \backslash frac\{y\_1\; -\; y\_0\}\{(x\_1\; -\; x\_0)^2\},$

:$b\_2\; =\; 3\; \backslash left(\backslash frac\{y\_1\; -\; y\_0\}\{\{(x\_1\; -\; x\_0)\}^2\}\; +\; \backslash frac\{y\_2\; -\; y\_1\}\{\{(x\_2\; -\; x\_1)\}^2\}\backslash right),$

:$b\_3\; =\; 3\; \backslash frac\{y\_2\; -\; y\_1\}\{(x\_2\; -\; x\_1)^2\}.$

For the three points

:$(-1,0.5),\backslash \; (0,0),\backslash \; (3,3),$

one gets that

:$k\_0\; =\; -0.6875,\backslash \; k\_1\; =\; -0.1250,\backslash \; k\_2\; =\; 1.5625,$

and from ({{EquationNote|10}}) and ({{EquationNote|11}}) that

:$a\_1\; =\; k\_0(x\_1\; -\; x\_0)\; -\; (y\_1\; -\; y\_0)\; =\; -0.1875,$

:$b\_1\; =\; -k\_1(x\_1\; -\; x\_0)\; +\; (y\_1\; -\; y\_0)\; =\; -0.3750,$

:$a\_2\; =\; k\_1(x\_2\; -\; x\_1)\; -\; (y\_2\; -\; y\_1)\; =\; -3.3750,$

:$b\_2\; =\; -k\_2(x\_2\; -\; x\_1)\; +\; (y\_2\; -\; y\_1)\; =\; -1.6875.$

In the figure, the spline function consisting of the two cubic polynomials $q\_1(x)$ and $q\_2(x)$ given by ({{EquationNote|9}}) is displayed.

• Akima spline
• Circular interpolation
• Cubic Hermite spline
• Centripetal Catmull–Rom spline
• Discrete spline interpolation
• Monotone cubic interpolation
• Non-uniform rational B-spline
• Multivariate interpolation
• Polynomial interpolation
• Smoothing spline
• Spline wavelet
• Thin plate spline
• Polyharmonic spline

{{Reflist}}

• {{cite journal |last=Schoenberg |first=Isaac J. |title=Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions:Part A.—On the Problem of Smoothing or Graduation. A First Class of Analytic Approximation Formulae |journal=Quarterly of Applied Mathematics |volume=4 |issue=2 |pages=45–99 |year=1946 |doi=10.1090/qam/15914 |doi-access=free }}
• {{cite journal |last=Schoenberg |first=Isaac J. |title=Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions:Part B.—On the Problem of Osculatory Interpolation. A Second Class of Analytic Approximation Formulae |journal=Quarterly of Applied Mathematics |volume=4 |issue=2 |pages=112–141 |year=1946 |doi=10.1090/qam/16705 |doi-access=free }}

• [http://tools.timodenk.com/?p=cubic-spline-interpolation Cubic Spline Interpolation Online Calculation and Visualization Tool (with JavaScript source code)]
• {{springer|title=Spline interpolation|id=p/s086820}}
• [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Cubic_spline_interpolation Dynamic cubic splines with JSXGraph]
• [https://www.youtube.com/view_play_list?p=DAB608CD1A9A0D55 Lectures on the theory and practice of spline interpolation]
• [https://web.archive.org/web/20090408054627/http://online.redwoods.cc.ca.us/instruct/darnold/laproj/Fall98/SkyMeg/Proj.PDF Paper which explains step by step how cubic spline interpolation is done, but only for equidistant knots.]
• [http://apps.nrbook.com/c/index.html Numerical Recipes in C, Go to Chapter 3 Section 3-3]
• [http://www.cs.tau.ac.il/~turkel/notes/numeng/spline_note.pdf A note on cubic splines]
• [https://websites.pmc.ucsc.edu/~fnimmo/eart290c_17/NumericalRecipesinF77.pdf Information about spline interpolation (including code in Fortran 77)]
• [https://github.com/msteinbeck/tinyspline TinySpline:Open source C-library for splines which implements cubic spline interpolation]
• [https://docs.scipy.org/doc/scipy/tutorial/interpolate.html SciPy Spline Interpolation:a Python package that implements interpolation]
• [https://github.com/ValexCorp/Cubic-Interpolation Cubic Interpolation:Open source C#-library for cubic spline interpolation]

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Category:Splines (mathematics)

Category:Interpolation