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Name

Synopsis

spline natural x y sx sy

spline quadratic x y sx sy

Description

Introduction

A spline is a device used in drafting to produce smoothed curves. The points of the curve, known as knots, are fixed and the spline, typically a thin strip of wood or metal, is bent around the knots to create the smoothed curve. Spline interpolation is the mathematical equivalent. The curves between adjacent knots are piecewise functions such that the resulting spline runs exactly through all the knots. The order and coefficients of the polynomial determine the "looseness" or "tightness" of the curve fit from the line segments formed by the knots.

The spline command performs spline interpolation using cubic ("natural") or quadratic polynomial functions. It computes the spline based upon the knots, which are given as x and y vectors. The interpolated new points are determined by another vector which represents the abscissas (x-coordinates) or the new points. The ordinates (y-coordinates) are interpolated using the spline and written to another vector.

Example

for {set i 10} {$i > 0} {incr i -1} {
set x($i-1) [expr $i*$i]
set y($i-1) [expr sin($i*$i*$i)]
}
We now have two vectors CWx and CWy representing the ten data points we’re trying to fit. The order of the values of CWx must be monotonically increasing. We can use the vector’s sort operation to sort the vectors.
x sort y
The components of CWx are sorted in increasing order. The components of CWy are rearranged so that the original x,y coordinate pairings are retained.

A third vector is needed to indicate the abscissas (x-coordinates) of the new points to be interpolated by the spline. Like the x vector, the vector of abscissas must be monotonically increasing. All the abscissas must lie between the first and last knots (x vector) forming the spline.

How the abscissas are picked is arbitrary. But if we are going to plot the spline, we will want to include the knots too. Since both the quadratic and natural splines preserve the knots (an abscissa from the x vector will always produce the corresponding ordinate from the y vector), we can simply make the new vector a superset of CWx. It will contain the same coordinates as CWx, but also the abscissas of the new points we want interpolated. A simple way is to use the vector’s populate operation.
x populate sx 10
This creates a new vector CWsx. It contains the abscissas of CWx, but in addition CWsx will have ten evenly distributed values between each abscissa. You can interpolate any points you wish, simply by setting the vector values.

Finally, we generate the ordinates (the images of the spline) using the spline command. The ordinates are stored in a fourth vector.
spline natural x y sx sy
This creates a new vector CWsy. It will have the same length as CWsx. The vectors CWsx and CWsy represent the smoothed curve which we can now plot.
graph .graph
.graph element create original -x x -y x -color blue
.graph element create spline -x sx -y sy -color red
table . .graph
The natural operation employs a cubic interpolant when forming the spline. In terms of the draftsmen’s spline, a natural spline requires the least amount of energy to bend the spline (strip of wood), while still passing through each knot. In mathematical terms, the second derivatives of the first and last points are zero.

Alternatively, you can generate a spline using the quadratic operation. Quadratic interpolation produces a spline which follows the line segments of the data points much more closely.
spline quadratic x y sx sy


Operations

spline natural x y sx sy

Computes a cubic spline from the data points represented by the vectors x and y and interpolates new points using vector sx as the x-coordinates. The resulting y-coordinates are written to a new vector sy. The vectors x and y must be the same length and contain at least three components. The order of the components of x must be monotonically increasing. Sx is the vector containing the x-coordinates of the points to be interpolated. No component of sx can be less than first component of x or greater than the last component. The order of the components of sx must be monotonically increasing. Sy is the name of the vector where the calculated y-coordinates will be stored. If sy does not already exist, a new vector will be created.

spline quadratic x y sx sy

Computes a quadratic spline from the data points represented by the vectors x and y and interpolates new points using vector sx as the x-coordinates. The resulting y-coordinates are written to a new vector sy. The vectors x and y must be the same length and contain at least three components. The order of the components of x must be monotonically increasing. Sx is the vector containing the x-coordinates of the points to be interpolated. No component of sx can be less than first component of x or greater than the last component. The order of the components of sx must be monotonically increasing. Sy is the name of the vector where the calculated y-coordinates are stored. If sy does not already exist, a new vector will be created.

References

Numerical Analysis
by R. Burden, J. Faires and A. Reynolds.    
Prindle, Weber & Schmidt, 1981, pp. 112

Shape Preserving Quadratic Splines 
by D.F.Mcallister & J.A.Roulier
Coded by S.L.Dodd & M.Roulier N.C.State University.

Keywords

Table of Contents

• Name
• Synopsis
• Description
• Introduction
• Example
• Operations
• References
• Keywords