Interpolation function

Question

I am facing a problem in getting a smooth plot of following data.

DD := {{{0, 0}, 1}, {{0, 0.1}, 1}, {{0, 0.2}, 1}, {{0, 0.3}, 
1}, {{0, 0.4}, 1}, {{0, 0.5}, 1}, {{0, 0.6}, 1}, {{0, 0.7}, 
1}, {{0, 0.736}, 1}, {{0.2, 0.0}, 0.997978}, {{0.2, 0.1}, 
0.99592}, {{0.2, 0.2}, 0.994118}, {{0.2, 0.3}, 
0.99321}, {{0.2, 0.4}, 0.990521}, {{0.2, 0.5}, 
0.990098}, {{0.2, 0.6}, 0.981427}, {{0.2, 0.684}, 
0.954755}, {{0.3, 0}, 0.99357}, {{0.3, 0.3}, 
0.985479}, {{0.3, 0.628105}, 0.927041}, {{0.4, 0}, 
0.991344}, {{0.4, 0.1}, 0.988842}, {{0.4, 0.3}, 
0.980593}, {{0.4, 0.4}, 0.972082}, {{0.4, 0.5573}, 
0.900049}, {{0.5, 0.0}, 0.98288}, {{0.5, 0.1}, 
0.979876}, {{0.5, 0.2}, 0.972208}, {{0.5, 0.3}, 
0.964005}, {{0.5, 0.4}, 0.943466}, {{0.5, 0.465}, 
0.914242}, {{0.6, 0}, 0.976438}, {{0.6, 0.1}, 
0.967633}, {{0.6, 0.2}, 0.960438}, {{0.6, 0.38848}, 
0.876153}, {{0.7, 0.0}, 0.96334}, {{0.7, 0.1}, 
0.953086}, {{0.7, 0.2}, 0.935014}, {{0.7, 0.2952}, 
0.874474}, {{0.8, 0}, 0.952486}, {{0.8, 0.1}, 
0.933406}, {{0.8, 0.198}, 0.874677}, {{0.9, 0}, 
0.928887}, {{0.9, 0.09983}, 0.866017}}

f = Interpolation[DD, InterpolationOrder -> 16];
ContourPlot[Quiet[Check[f[a, g], Null]], {a, 0, 0.9}, {g, 0, 0.75}, 
PlotLegends -> Automatic, MaxRecursion -> 5]

I am getting a rough plot on using Mathematica 9. I am looking to get a smooth plot. Any suggestions are welcome.

Did any of the answers satisfied your need? There are things to do after your question is answered. It's a good idea to stay vigilant for some time, better approaches may come later improving over previous replies. Experienced users may point alternatives, caveats or limitations. New users should test answers before voting and wait 24 hours before accepting the best one. One weeks is enough wait. Participation is essential for the site, please do your part. — rhermans, Jul 11 '18 at 18:24
@rhermans. Yes, suggestions for this problem worked for me. Many thanks to all. — , Jul 13 '18 at 16:22

Coolwater · Answer 1 · 2018-07-05T17:04:57.633

At least in version 11.3 when Interpolation is called there is the error

Interpolation::udeg: Interpolation on unstructured grids is currently only supported for InterpolationOrder->1 or InterpolationOrder->All. Order will be reduced to 1.

Using InterpolationOrder -> All and appropriate PlotRange fixes the plot:

DD = {{{0,0},1},{{0,0.1},1},{{0,0.2},1},{{0,0.3},1},{{0,0.4},1},{{0,0.5},1},
      {{0,0.6},1},{{0,0.7},1},{{0,0.736},1},{{0.2,0.0},0.997978},{{0.2,0.1},0.99592},
      {{0.2,0.2},0.994118},{{0.2,0.3},0.99321},{{0.2,0.4},0.990521},{{0.2,0.5},0.990098},
      {{0.2,0.6},0.981427},{{0.2,0.684},0.954755},{{0.3,0},0.99357},{{0.3,0.3},0.985479},
      {{0.3,0.628105},0.927041},{{0.4,0},0.991344},{{0.4,0.1},0.988842},{{0.4,0.3},0.980593},
      {{0.4,0.4},0.972082},{{0.4,0.5573},0.900049},{{0.5,0.0},0.98288},{{0.5,0.1},0.979876},
      {{0.5,0.2},0.972208},{{0.5,0.3},0.964005},{{0.5,0.4},0.943466},{{0.5,0.465},0.914242},
      {{0.6,0},0.976438},{{0.6,0.1},0.967633},{{0.6,0.2},0.960438},{{0.6,0.38848},0.876153},
      {{0.7,0.0},0.96334},{{0.7,0.1},0.953086},{{0.7,0.2},0.935014},{{0.7,0.2952},0.874474},
      {{0.8,0},0.952486},{{0.8,0.1},0.933406},{{0.8,0.198},0.874677},{{0.9,0},0.928887},
      {{0.9,0.09983},0.866017}};

f = Interpolation[DD, InterpolationOrder -> All];

ContourPlot[f[x, y], {x, 0, 0.9}, {y, 0, 0.75}, PlotLegends -> Automatic, PlotPoints -> 50,
              PlotRange -> {All, All, {-0.4, 1.4}}, Contours -> Range[-4/10, 12/10, 1/5]]

In V10 ConvexHullMesh can be used to specify the plot region (because your data supports every edge of its convex hull):

ContourPlot[f[x, y], {x, y} ∈ ConvexHullMesh[DD[[All, 1]]], 
  PlotLegends -> Automatic, PlotPoints -> 50, PlotRange -> All]

Edit: In V9 you can use the order 1 interpolation as a region function:

fDom = Interpolation[DD, InterpolationOrder -> 1];
f = Interpolation[DD, InterpolationOrder -> All];

ContourPlot[If[Check[fDom[a, g], True] =!= True, f[a, g], Null], {a, 0, 0.9}, {g, 0, 0.75},
         PlotLegends -> Automatic, MaxRecursion -> 1, PlotPoints -> 50, PlotRange -> All]

In 11.3 the output is practically the same as that of the ConvexHullMesh version.

Edit: Maybe using a fit is better:

fit[x_, y_] = Normal[LinearModelFit[Append @@@ DD, x^# y^#2 & @@@ Tuples[{0, 1, 2, 3, 4}, 2], {x, y}]];
fDom = Interpolation[DD, InterpolationOrder -> 1];
ContourPlot[If[Check[fDom[a, g], True] =!= True, fit[a, g], Null], {a, 0, 0.9}, {g, 0, 0.75},
   PlotRange -> {All, All, {0.86, 1.02}},
   PlotLegends -> Automatic, PlotPoints -> 50, MaxRecursion -> 1,
   Contours -> (Rescale[#, {Min[#], Max[#]}, {0.84, 1}] &[Range[20] // Sqrt])]

Thanks for suggestions..Actually I am looking to get smoother version of plot that I show. I checked interpolation order is not changing the smoothness except interpolationorder->All. The plots that you are showing are not surving my purpose. Thanks again. — , Jul 05 '18 at 16:28
Thanks. In fDom function why we can not have interpolation order higher than 1. — , Jul 13 '18 at 16:25

Anton Antonov · Answer 2 · 2018-07-04T15:52:59.910

I am using Version 11.3, but I hope the code below would produce similar results for you in Version 9.

dd = Flatten /@ {{{0, 0}, 1}, {{0, 0.1}, 1}, {{0, 0.2}, 1}, {{0, 0.3},
      1}, {{0, 0.4}, 1}, {{0, 0.5}, 1}, {{0, 0.6}, 1}, {{0, 0.7}, 
     1}, {{0, 0.736}, 1}, {{0.2, 0.0}, 0.997978}, {{0.2, 0.1}, 
     0.99592}, {{0.2, 0.2}, 0.994118}, {{0.2, 0.3}, 
     0.99321}, {{0.2, 0.4}, 0.990521}, {{0.2, 0.5}, 
     0.990098}, {{0.2, 0.6}, 0.981427}, {{0.2, 0.684}, 
     0.954755}, {{0.3, 0}, 0.99357}, {{0.3, 0.3}, 
     0.985479}, {{0.3, 0.628105}, 0.927041}, {{0.4, 0}, 
     0.991344}, {{0.4, 0.1}, 0.988842}, {{0.4, 0.3}, 
     0.980593}, {{0.4, 0.4}, 0.972082}, {{0.4, 0.5573}, 
     0.900049}, {{0.5, 0.0}, 0.98288}, {{0.5, 0.1}, 
     0.979876}, {{0.5, 0.2}, 0.972208}, {{0.5, 0.3}, 
     0.964005}, {{0.5, 0.4}, 0.943466}, {{0.5, 0.465}, 
     0.914242}, {{0.6, 0}, 0.976438}, {{0.6, 0.1}, 
     0.967633}, {{0.6, 0.2}, 0.960438}, {{0.6, 0.38848}, 
     0.876153}, {{0.7, 0.0}, 0.96334}, {{0.7, 0.1}, 
     0.953086}, {{0.7, 0.2}, 0.935014}, {{0.7, 0.2952}, 
     0.874474}, {{0.8, 0}, 0.952486}, {{0.8, 0.1}, 
     0.933406}, {{0.8, 0.198}, 0.874677}, {{0.9, 0}, 
     0.928887}, {{0.9, 0.09983}, 0.866017}};

ListPointPlot3D[dd, PlotRange -> All]

At this point I try to device some function for ContourPlot's option RegionPlot. I am using Internal`ListMin from this discussion: "upper envelope of data". (I assume the function ListMin is also available in Version 9.)

upper = -Internal`ListMin[-Map[Most, dd]];
ListPlot[upper]

fb = Interpolation[upper, InterpolationOrder -> 2];

f = Interpolation[dd, InterpolationOrder -> All];

ContourPlot[f[x, y], {x, 0, 0.9}, {y, 0, 0.75}, 
 PlotLegends -> Automatic, PlotPoints -> 120, 
 PerformanceGoal -> "Speed", PlotRange -> All, 
 RegionFunction -> Function[{x, y}, 0 <= y <= fb[x]]]

Interpolation function

2 Answers2