Extending a plot over a secondary variable

Question

Apologies for the vague title; couldn't really think of a better way of putting it.

I have a function $F$ of two variables, $\rho\in[0,\infty)$ and $f\in[0,1]$, that I wish to create a plot over (e.g. with CountourPlot or DensityPlot). I can numerically generate the entire curve of $F$ over $f$ for a particular value of $\rho$ relatively cheaply, but I can't directly compute $F$ for individual values of $f$.

Is there an efficient way of taking such a curve and plotting it for a range of $\rho$ without Mathematica having to evaluate the function independently at every $(\rho,f)$?

---

Here's how I compute $F$:

Lorentz[data_] := Module[{xs, ys},
   xs = Range[0, Length[data]];
   ys = Prepend[Accumulate[Sort[data]], 0];
   Transpose[{xs/Last@xs, ys/Last@ys}]];

L = Interpolation[Lorentz[data], InterpolationOrder -> 1];
F = 1 - L[1 - #] &;

data is a list of real numbers generated according to the value of $\rho$.

Essentially, I generate a Lorentz curve numerically as a list of $(x,y)$ tuples, turn it into a function via Interpolation, reflect it a couple of times, and that's my function $F$.

Answer 1

Lets change the definition of your function a bit. here n is the number of points you take in the direction of variable f.

Lorentz[rho_, n_: 100] := Module[{data, xs, ys},
  data = RandomVariate[NormalDistribution[1, 3], n];
  xs = Range[0, n];
  ys = Prepend[Accumulate[Sort[data]], 0];
  xs /= Last@xs;
  ys /= Last@ys;
  N@Transpose@{Transpose@{xs, ConstantArray[rho, n + 1]}, ys}
  ];
(* Discretize f direction with 1000 *)
(* Discretize rho direction say {0,5} with 0.1 *)
L = Interpolation[Flatten[Table[Lorentz[i, 1000], {i, 0, 5, .1}], 1],
InterpolationOrder -> 3];
F[f_, rho_] := 1 - L[1 - f, rho];

Now lets plot what you want!

GraphicsGrid[{{Plot3D[F[f, rho], {f, 0, 1}, {rho, 0, 5}, 
Mesh -> False, PlotPoints -> 100, PlotRange -> All, 
Boxed -> False,ColorFunction -> "Rainbow"], 
ContourPlot[F[f, rho], {f, 0, 1}, {rho, 0, 5}, Mesh -> None, 
PlotPoints -> 100, PlotRange -> All,ColorFunction -> "Rainbow",
ContourLabels -> True]}}, ImageSize -> 800]

The plot takes some time to evaluate. For fast execution you can reduce the PlotPoints.