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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.

BlacKow
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    The third plot only looks messed up because you're using too few PlotPoints (or too large a $xy$ range). You're correct in that ColorFunctionScaling is the issue for the first two plots, they are scaled differently, as is clear from the documentation. – Roman Apr 30 '19 at 17:12
  • @Roman Why are the first two plots scaled differently? They both have ColorFunctionScaling true. Also z=xy so Hue[z] should be same as Hue[xy] – BlacKow Apr 30 '19 at 18:44
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    In the first plot, both $x$ and $y$ are scaled to the range $[0,1]$ and thus $x y$ (meaning the product of scaled-$x$ and scaled-$y$) is zero in the lower-left corner, left edge, and lower edge, and 1 in the upper-right corner. In the second plot, $z$ is scaled to the range $[0,1]$ and thus zero in the lower-right and upper-left corners, and 1 in the lower-left and upper-right corners. If you play with Hue@x, Hue@y, and Hue@z both with and without ColorFunctionScaling it all becomes clear. – Roman Apr 30 '19 at 18:53
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    From the documentation: "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." Meaning that every argument of the color function will be scaled to $[0,1]$. – Roman Apr 30 '19 at 19:04
  • @Roman OK, I see it now.. somehow I wrongfully assumed that the ColorFunctionScaling controls the scaling of the value, not arguments. Honestly I'm not sure what's the point of scaling arguments to [0,1]. If you make it an answer I will gladly accept. Thanks! – BlacKow Apr 30 '19 at 19:08

1 Answers1

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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.

Roman
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  • Increasing PlotPoints make it smoother, but you need to go with manually scaled ColorFunction -> Function[{x, y, z}, Hue[0.01 x y]] to make a desired effect. – BlacKow Apr 30 '19 at 19:22