Plot a function on a 2D surface

Question

My Question

How do you plot a function $f(x,y)=z$ on a 2D surface $g(x,y)=z$.

A Simple Example

f[x_, y_] := (x - 0.5)^2 + (y - 0.5)^2;
g[x_, y_] := y Sin[x y]
DensityPlot[f[x, y], {x, 0, 2}, {y, 0, 2}]
(* 2D surface I want to plot on *)
Plot3D[g[x, y], {x, 0, 2}, {y, 0, 2}]

How can I plot the first on the second?

Notes

• If you have any questions or need clarification please ask.

Answer 1

ColorFunction

Consider:

Plot3D[g[x, y], {x, 0, 2}, {y, 0, 2}, 
 ColorFunction -> 
  Function[{x, y, z}, ColorData["RustTones"][f[x, y]/4.5]], 
 ColorFunctionScaling -> False]

Replace "RustTones" with your favorite value from ColorData["Gradients"] (I do not know how to mimic the original DensityPlot color scheme).

The DensityPlot output + PlotStyle

Alternatively, use the original DensityPlot as a texture:

img = DensityPlot[f[x, y], {x, 0, 2}, {y, 0, 2}, Frame -> False, 
  PlotRangePadding -> None]
Plot3D[g[x, y], {x, 0, 2}, {y, 0, 2}, PlotStyle -> Texture[img]]

Answer 2

You can also use a custom color function using your function f and the default color function for DensityPlot (which is "M10DefaultDensityGradient") as follows:

minmax = Through[{NMinValue, NMaxValue}[{f[x, y], 0 <= x <= 2 && 0 <= y <= 2}, {x, y}]];

cF1 = ColorData["M10DefaultDensityGradient"][Rescale[f[#, #2], minmax]] &;

Plot3D[g[x, y], {x, 0, 2}, {y, 0, 2}, ColorFunction -> cF1,
 ColorFunctionScaling -> False, Lighting -> "Ambient", ViewPoint -> {1.5, -.5, 3}]

Use a different gradient color scheme, say "CMYKColors", instead of "M10DefaultDensityGradient" to get: