NAME¶
spline - Fit curves with spline interpolation
SYNOPSIS¶
spline natural x y sx sy
spline quadratic x y sx sy
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 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¶
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.
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.
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.
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 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.
The original code for the quadratic spline can be found in TOMS #574.
KEYWORDS¶
spline, vector, graph
