Maximum and minimum plotted on a curve

Question

Through code:

f[{x_, y_, z_}] = x^4 + y^4 + z^4;

r[t_] = {1/3 + 1/3 Cos[t] - 1/Sqrt[3] Sin[t],
         1/3 + 1/3 Cos[t] + 1/Sqrt[3] Sin[t],
         1/3 - 2/3 Cos[t]};
{min, max} = {0, 2 Pi};

ParametricPlot3D[
 r[t],
 {t, min, max},
 ColorFunction -> Function[t, 
   ColorData["Rainbow"][f[r[t]]]],
 AxesLabel -> {x, y, z},
 BoxRatios -> {1, 1, 1},
 PlotLegends -> BarLegend[{"Rainbow",
    {MinValue[{f[r[t]], min <= t <= max}, t],
     MaxValue[{f[r[t]], min <= t <= max}, t]}}]]

I get:

which it is certainly not what you want, because through this other code:

f[x_, y_, z_] = x^4 + y^4 + z^4;

A = ImplicitRegion[x^2 + y^2 + z^2 <= 1 && x + y + z <= 1, {x, y, z}];

SliceDensityPlot3D[
 f[x, y, z],
 BoundaryDiscretizeRegion[A],
 {x, y, z} \[Element] DiscretizeRegion[A],
 ColorFunction -> "Rainbow",
 AxesLabel -> Automatic,
 PlotLegends -> Automatic]

I get:

where the colors are very clear that it must be expected on the circumference above.

Where am I wrong?

Answer 1

By default, color data is scaled such that it uses the entire range of the applied color function. To avoid this, set ColorFunctionScaling->False. Furthermore, ColorFunction, according to the documentation, should be a function of x,y,z,u (where u is t in your code). So here is the full call:

ParametricPlot3D[r[t], {t, min, max}, 
 ColorFunction -> 
  Function[{x, y, z, t}, ColorData["Rainbow"][f[r[t]]]],
 ColorFunctionScaling -> False, AxesLabel -> {x, y, z}, 
 BoxRatios -> {1, 1, 1}, 
 PlotLegends -> 
  BarLegend[{"Rainbow", {MinValue[{f[r[t]], min <= t <= max}, t], 
     MaxValue[{f[r[t]], min <= t <= max}, t]}}]]