# Produce a spline from a set of {{x, y}, z} points and get its parameters/expression

I have produced a set of points in a variable of the type {{x1, y1}, z1},{{x2, y2}, z2}, ,... }, where {x, y} should be the arguments of the function and z is the associated value. What I need to obtain is the continuous function in order to generate z for any value of {x, y} in the domain implicit in the data.

As I understand it, a good way to do this would be to use a 2D Spline. At first I had an unstructured grid, but with the help of the community, now I've produced a grid and values of this form (the real list has been "thinned"):

{{{0.11, 0.1}, 0.621687}, {{0.11, 0.3}, 0.674455}, {{0.11, 0.5}, 0.743462},
{{0.11, 0.7}, 0.803179}, {{0.11, 0.9}, 0.843444}, {{0.11, 1.1}, 0.866607},
{{0.11, 1.3}, 0.881172}, {{0.11, 1.5}, 0.890488}, {{0.11, 1.7}, 0.896887},
{{0.11, 1.9}, 0.901538}, {{0.11, 2.1}, 0.905067}, {{0.11, 2.3}, 0.907836},
{{0.11, 2.5}, 0.910066}, {{0.11, 2.7}, 0.911902}, {{0.11, 2.9}, 0.913404},
(*==============delimiter==================*)
{{0.15, 0.1}, 0.15}, {{0.15, 0.3}, 0.15}, {{0.15, 0.5}, 0.15},
{{0.15,0.7}, 0.414754}, {{0.15, 0.9}, 0.648793}, {{0.15, 1.1}, 0.768185},
{{0.15, 1.3}, 0.832127}, {{0.15, 1.5}, 0.862517}, {{0.15, 1.7}, 0.879246},
{{0.15, 1.9}, 0.889989}, {{0.15, 2.1}, 0.897541}, {{0.15, 2.3}, 0.90316},
{{0.15, 2.5}, 0.907512}, {{0.15, 2.7}, 0.910985}, {{0.15, 2.9}, 0.913761},
(*==============delimiter==================*)
{{0.19, 0.1}, 0.19}, {{0.19, 0.3}, 0.19}, {{0.19, 0.5}, 0.19},
{{0.19,0.7}, 0.19}, {{0.19, 0.9}, 0.402054}, {{0.19, 1.1}, 0.658819},
{{0.19, 1.3}, 0.81877}, {{0.19, 1.5}, 0.871256}, {{0.19, 1.7}, 0.88874},
{{0.19, 1.9}, 0.898598}, {{0.19, 2.1}, 0.905459}, {{0.19, 2.3}, 0.910567},
{{0.19, 2.5}, 0.914527}, {{0.19, 2.7}, 0.917694}, {{0.19, 2.9}, 0.920228}}


Further, I use:

fspl = Interpolation[Xac, Method -> "Spline", InterpolationOrder -> 2]


and obtain, in principle, satisfying results if I could get the expression for the spline (for use outside Mathematica). Searching on stackexchange shows that people have had similar problems, but I found no evident solution. I tried to use fspl["Methods"], but the list it returns does not seem to be really helpful.

• Look up Interpolation[]. – J. M.'s torpor Mar 15 '16 at 13:00
• After I've transformed the data to {{x1,y1},z1},{{x2,y2},z2},... and use Interpolation I get the error "There are duplicated abscissa points in"... But I intend to have duplicate x or y values... – user14679 Mar 15 '16 at 13:37
• @user14679 - but it wouldn't make sense for there to be exactly duplicated {x, y} pairs with different z values. Also, Interpolation will not give you the form of the interpolating function, it will just allow you to get values in between your grid points. To actually get the spline function is a nontrivial task. – Jason B. Mar 15 '16 at 14:03
• Oh, but my real aim is to get the spline function expression, because I will further use it out of Mathematica. – user14679 Mar 15 '16 at 14:06
• You do have equal $\{x,y\}$ pairs. With the above data assigned to dat then Select[# > 1 &]@Counts[Most /@ dat] selects one such duplicate pair. – Edmund Mar 15 '16 at 14:23

I have done this before in some previous spline-related threads, but here it is again:

fspl = Interpolation[Xac, Method -> "Spline", InterpolationOrder -> 2];

fb = First[Cases[fspl, _BSplineFunction, ∞]];
{sd, scpts, sk} = fb /@ {"Degree", "ControlPoints", "Knots"};

(* explicitly constructed B-spline surface *)
ip[x_, y_] = Fold[#2.#1 &, scpts,
MapThread[Table[BSplineBasis[{#1, #2}, k - 1, #3], {k, #4}] &,
{sd, sk, {x, y}, Dimensions[scpts]}]];


Compare:

{fspl[0.16, 2.3], ip[0.16, 2.3]}
{0.9038789687499998, 0.90387896875}