ColorFunction causing division by zero error (going outside domain?) [duplicate]

Question

On the following plot example, I get a division by zero error (10.4.1):

ListPointPlot3D[{{1, 1, 1}, {2, 2, 2}}, ColorFunction -> Function[{x, y, z}, ColorData["TemperatureMap"][1/x]]]
(*Power::infy: Infinite expression 1/0. encountered. >>*)

while there seems to be no problem with the ColorFunction:

Function[{x, y, z}, ColorData["TemperatureMap"][1/x]] @@@ {{1, 1, 1}, {2, 2, 2}}

For what I could get, this is true also for other type of plots.

We can try to do the scaling ourselves (not that with the way it is defined this should matter).

MinMax[1/#[[1]] & /@ {{1, 1, 1}, {2, 2, 2}}]
(*{1/2, 1}*)

Rescale[1/#[[1]], MinMax[{1/2, 1}]] & /@ {{1, 1, 1}, {2, 2, 2}}
(*{1, 0}*)

But it works:

ListPointPlot3D[{{1, 1, 1}, {2, 2, 2}}, 
 ColorFunction -> Function[{x, y, z}, ColorData["TemperatureMap"][
 Rescale[1/x, MinMax[1/#[[1]] & /@ {{1, 1, 1}, {2, 2, 2}}]]]], 
 ColorFunctionScaling -> False]

What's going on? It seems that Mathematica is defining the color scale outside the plot range. If so, why?

Answer 1

This is expected behavior since the values are all rescaled to lie between 0 and 1 prior to feeding them to the ColorFunction. So there will always be some portion of the plotted region that corresponds to a value of 0 and a region that corresponds to a value of 1.

Look at this simplified example,

DensityPlot[2 + UnitStep[x], {x, -3, 3}, {y, -3, 3}, 
 ColorFunction -> (Which[# == 0, Black, # == 1, White, True, Red] &)]

The function value is either 2 or 3 at every point in the range, but the ColorFunction only sees values of either 0 or 1.

You need to turn ColorFunctionScaling to False in order for the colors to work the way you want them to

ListPointPlot3D[Array[{#, #, #} &, 10], 
   ColorFunction -> 
    Function[{x, y, z}, ColorData["TemperatureMap"][1/x]],
   ColorFunctionScaling -> #] & /@ {True, False}

Notice you get the right colors here,

ColorData["TemperatureMap"][1/#] & /@ Range[10]