(* Exploring splines *)


(* To run on command line:  math -run "<<spline.math" *)
(* The above line assumes that math kernel is aliased to "math" *)
(* For example, on my machine alias
math='/Applications/Mathematica.app/Contents/MacOS/MathKernel' *)
(* or simply use << spline.math once math kernel is already running *)

<<JavaGraphics`


(*
<<NumericalMath`SplineFit` (* First, load the package. Seems deprecated,
keep commented *)
*)

Needs["Splines`"]

F[x_] := 1/(1 + 25*x*x); (* define Runge's function *) 


a = -1.0; b= 1.0; (* define the interpolating interval *) 

plotfunction = Plot[F[x], {x, a, b}, PlotStyle->{Hue[1], Thickness[0.01]}, PlotLabel->"Original function", DisplayFunction -> Identity ]

n=16; h = (b-a)/n;  (* define knots *)



data = Table[N[{x, F[x]}], {x, a, b, h}] (* produce data points at each knot *)

 
 

(* Uncomment below to try spline fit for a parametric curve *)


(*  

data = Table[N[{Sin[t], Cos[3t]}], {t, a, b, h}] (* parametric curve *)

*) 

plotfunction = ParametricPlot[{Sin[t], Cos[3t]}, {t, a, b}, PlotStyle->{Hue[1], Thickness[0.005]}, PlotLabel->"A parametric curve" ] 

  
 

plotdata = ListPlot[data, PlotLabel -> "Data points", Prolog-> AbsolutePointSize
[5] ]


fit = SplineFit[data, Cubic] (* Do the Cubic spline interpolation *)


plotfit = ParametricPlot[fit[t], {t, 0, n}, PlotStyle->{Hue[0.5], Thickness[0.005]}, PlotLabel-> "Cubic spline", DisplayFunction -> Identity ]

(* Show all graphs at once *) 
Show[plotfunction, plotdata, plotfit, PlotLabel -> FontForm["red - original f(x), blue - Cubic Spline ", {"Courier", 15}], Prolog-> AbsolutePointSize[5], DisplayFunction -> $DisplayFunction ]  


(* Print the value of the spline at the leftmost knot. Output format {x,y} *)
Print[fit[0.0]]

(* And at a point just a bit to the right *)

Print[fit[0.01]]

Clear[data]

Clear[fit]