spline natural x y sx sy
spline quadratic x y sx sy
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.
# 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 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
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.