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man pages->Linux man pages -> spline (3)
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### Contents

```
spline(BLT 2.4) 					       spline(BLT 2.4)

______________________________________________________________________________
```

### NAME[Toc][Back]

```       spline -  Fit curves with spline interpolation
```

### SYNOPSIS[Toc][Back]

```       spline natural x y sx sy

spline quadratic x y sx sy
______________________________________________________________________________
```

### DESCRIPTION[Toc][Back]

```       The  spline  command  computes a spline fitting a set of data points (x
and y vectors) and produces a vector of	the  interpolated  images  (ycoordinates)
at a given set of x-coordinates.
```

### INTRODUCTION[Toc][Back]

```       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[Toc][Back]

```       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 popu-
late 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[Toc][Back]

```       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[Toc][Back]

```       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[Toc][Back]

```       spline, vector, graph

spline(BLT 2.4)
```
[ Back ]
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