Provided by: blt-dev_2.5.3+dfsg-8_amd64 
NAME
spline - Fit curves with spline interpolation
SYNOPSIS
spline natural x y sx sy
spline quadratic x y sx sy
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DESCRIPTION
The spline command computes a spline fitting a set of data points (x and y vectors) and produces a vector
of the interpolated images (y-coordinates) at a given set of x-coordinates.
INTRODUCTION
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.
EXAMPLE
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.
# 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)]
}
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.
x sort y
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.
x populate sx 10
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.
spline natural x y sx sy
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.
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 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.
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.
The original code for the quadratric spline can be found in TOMS #574.
KEYWORDS
spline, vector, graph
BLT 2.5 blt::spline(3tcl)