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plotting - Plot3D: ColorFunction depending on (x,y) or z



While trying to answer this question I found out "weird" behavior of ColorFunction



Plot3D[x y, {x, -10, 10}, {y, -10, 10}, PlotRange -> Full, 
 ColorFunction -> Function[{x, y, z}, Hue@(x y)]]

gives incorrect colors


enter image description here


the virtually identical code (from naive point of view)


Plot3D[x y, {x, -10, 10}, {y, -10, 10}, PlotRange -> Full, 
 ColorFunction -> Function[{x, y, z}, Hue@z]]

gives correct plot



enter image description here


Also if you disable ColorFunctionScaling as many posts (e.g. this) suggest


Plot3D[x y, {x, -10, 10}, {y, -10, 10}, PlotRange -> Full, 
 ColorFunction -> Function[{x, y, z}, Hue@(x y)], 
 ColorFunctionScaling -> False]

the colors are really messed up


enter image description here


So the question - why are the first two plots different? A bug?


I'm using Mathematica 11.1 on Linux.




Answer



As the documentation on ColorFunctionScaling says,



ColorFunctionScaling is an option for graphics functions that specifies whether arguments supplied to a color function should be scaled to lie between 0 and 1.



In the first plot, x and y are both scaled to the range [0,1]. This means that the ColorFunction does not receive x and y, but rather (x+10)/20 and (y+10)/20, and the color is then the hue of (x+10)(y+10)/400.


In the second plot, z is scaled to the range [0,1]. This means that the ColorFunction does not receive z, but rather (z+100)/200 as the third argument.


In the third plot, the number of PlotPoints is too small and thus you see Moiré effects.


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