Curve fitting has many applications. In graphs, curve fitting can
be useful for displaying curves which are aesthetically pleasing to the
eye. Another advantage is that you can quickly generate arbitrary points
on the curve from a small set of data points.
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 polynominal
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.

Before we can use the **spline** command, we need to create two BLT
vectors which will represent the knots (x and y coordinates) of the
data that we’re going to fit. Obviously, both vectors must be the
same length.
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# Create sample data of ten points.
vector x(10) y(10)

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)]
}

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We now have two vectors **x** and **y** representing the ten data
points we’re trying to fit. The order of the values of **x** must
be monotonically increasing. We can use the vector’s **sort** operation
to sort the vectors.

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x sort y

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The components of **x** are sorted in increasing order. The
components of **y** 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 **x**.
It will contain the same coordinates as **x**, but also the
abscissas of the new points we want interpolated. A simple way is to
use the vector’s **populate** operation.

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x populate sx 10

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This creates a new vector **sx**. It contains the abscissas of
**x**, but in addition **sx** 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.

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spline natural x y sx sy

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This creates a new vector **sy**. It will have the same length as
**sx**. The vectors **sx** and **sy** represent the smoothed
curve which we can now plot.

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graph .graph
.graph element create original -x x -y x -color blue
.graph element create spline -x sx -y sy -color red
table . .graph

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The **natural** operation employs a cubic interpolant when forming
the spline. In terms of the draftmen’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.

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spline quadratic x y sx sy

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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.

The original code for the quadratric spline can be found in TOMS #574.