color - Create colour with given lightness value

How can I create a colour of my chosen hue with a given fixed lightness level?

For example, how can I create a reddish colour, col, so that ColorConvert[col, "GrayScale"] will return precisely GrayLevel[0.6]?

Does Mathematica have something built-in to assist in this, or do I need to read up on how colours are converted between various representations? Note that the b value in Hue[h, s, b] does not correspond to the GrayLevel value. I need to have precise control over what certain colours I use will look like when converted to greyscale.

Answer

Conversion Formula

ColorConvert uses the following formula for "Grayscale" conversion:

$ \mathrm{Grayscale} = 0.299 R + 0.587 G + 0.114 B$

where $R$, $G$, and $B$ are normalized.

Interactive Example

The following manipulate example will help you finding (and confirming) the conversion based on a fixed $\mathrm{Grayscale}$ and $R$ values.

f[g_, gs_, r_] := (gs - .299 r - .587 g)/.114;

Manipulate[
 Column[{
   Show[
    Graphics[{Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}},
       VertexColors -> {RGBColor[r, 0, 0], RGBColor[r, 1, 0], 
         RGBColor[r, 1, 1], RGBColor[r, 0, 1]}]},
     PlotRange -> {{0, 1}, {0, 1}}, PlotRangeClipping -> True, 
     AspectRatio -> 1,

     ImageSize -> 400,
     Frame -> True, FrameLabel -> {"Green", "Blue"},
     LabelStyle -> {FontFamily -> "Arial", FontSize -> 14}],
    Plot[f[g, gs, r], {g, 0, 1}],
    Graphics[{
      Locator[
       Dynamic[pt, (With[{b = f[#[[1]], gs, r]}, 
           If[0 <= b <= 1, pt = {#1[[1]], b}, {.5, .5}]]) &]]
      }]
    ],

   "",
   Row[{"ColorConvert[RGBColor[", Prepend[pt, r], 
     ",\"Grayscale\"]==\n", 
     ColorConvert[RGBColor[Prepend[pt, r]], "Grayscale"]}]
   }],
 {{pt, {.5, .5}}, ControlType -> None},
 {{gs, .5, "Grayscale"}, 0, 1, Appearance -> "Labeled"},
 {{r, 1, "Red"}, 0, 1, Appearance -> "Labeled"},
 SaveDefinitions -> True]

The blue line indicates the colors which has the exact grayscale value. If you move the locator, it tries to snap in to the line and show you RGB value as well as converted graysale value (which should be the same as the slider value).

Discussion

belisarius asked, "Then what is the yellowest color whose grayscale value is .3?".

There are several problems. First, what is "yellow"-ish colors? Are these colors close to {1,1,0} in RGB space? Then the solution is the point on the plane 0.299 x + 0.587 y + 0.114 z = 0.3 whose distance is the shortest from {1,1,0} which can be easily solvable. But you will quickly realize that it is not really "yellow". For instance, {1,0,0} and {0,1,0} are both exactly the same distance from the yellow, but none perceive them as yellowish. Then, is it a point with the proportion of R and G is 1:1? It is better but not entirely. Red and green have difference luminescence (thus different weights in grayscale conversion). Now, should we consider the weight? Well, then how about gamma and gamut? Then Euclidean distance becomes useless... I wouldn't even start the fact that human color perception is not linear but more of quadratic space... And then there is a whole story about chromaticity and visible spectrum (then different "yellow" there).

Getting the closest color for some color can be surprisingly hard.