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The "bicone" representation of the RGB color space

Doug Kerr

Well-known member
We often see reference made to the fact that the RGB color space can be understood in terms of a "bicone" in three-dimensional space - a figure comprising two cylindrical cones with their circular bases joined together.

Those putting forth this representation rarely describe what the three coordinates of this figure represent. But if pressed, they generally suggest that (and here visualize the solid with its axis - running between its two "points" - vertical) that the vertical coordinate represents luminance (or something like it), the "radial" coordinate (distance from the axis in a horizontal plane ) represents saturation (or something like it), and the "azimuthal" coordinate (angular, like longitude) represents hue (in the familiar "color wheel" paradigm).

The reason that these coordinates are not actual luminance and saturation is that with those the figure would be gravely asymmetrical, and wouldn't have a nice "equator". (This results from several facts, mostly the greatly differing impact of the three primaries, R, G, and B, on actual luminance, along with the untidy matter of defining saturation.)

Why does this figure shrink in radius to just a point at the top and the bottom?

Well, regarding the top half, the concept is that, as we increase in luminance, the range of saturation we can attain in an RGB color space diminishes, until at the highest achievable luminance (where RGB=255,255,255), there can be only one chromaticity (white), and its saturation is zero.

Even though the vertical coordinate is not (real) luminance, and the radius not (real) saturation, this notion is fair enough as a "metaphor".

Now for the lower half. The fact that the cross-section of the figure shrinks as we go "below the equator" is supposed to illustrate the supposed fact that, as the luminance declines, the range of saturation decreases, eventually reaching zero at the bottom (zero luminance), where the only color we can have is black, which "clearly has zero saturation".

Well, in fact, for zero luminance, saturation and hue are undefined, not zero. And if we go just an infinitesimal amount up (to a very tiny luminance), we have available the full range of hue and saturation implied by the "low luminance chromaticity gamut" of an RGB color space - the gamut bounded by the triangle joining the chromaticities of the three primaries, R, G,. and B.

So, if we accept that the actual coordinates of this figure are "(per)versions" of actual luminance and saturation that allow us to avoid the asymmetry of the actual figure (again owing to the differing impact of the three primaries on luminance), then we would find that this metaphorical solid would comprise a cone atop a cylinder. The cone is not "fair enough" as a metaphor for the Southern hemisphere.

(The actual shape of a luminance-hue-saturation plot of an RGB gamut is quite a different shape altogether - and recall, as I recently discussed in another note, there is not a unique quantitative definition of saturation we could comfortably use for that.)

Now, what might legitimately lead to the "bicone" metaphor? Well, if we use for the radial coordinate the property of chroma (the "magnitude" of chrominance), and warp it a bit so as to help the figure come out as symmetrical, we would get a figure whose lower half was a cone but whose upper half was a "cone with sunken sides" (like the blast diverter under a rocket test stand).

The HSB color space

Now what about the infamous HSB color space? Does it lead to a "bicone" representation?

Note that there are many flavors of this, which complicate the issue. Let's assume the Photoshop flavor. Its three coordinates are:

H ("hue") - a reasonable representation of hue as an angle, based on the familiar "color wheel" paradigm. It runs from 0° to 360°

S ("saturation") - a value that is something like saturation, although to give it a simple definition, it does not accurately reflect actual saturation (under any of the credible definitions). It runs from 0% to 100%.

B ("brightness") - a value that has a vague relationship to luminance. It is defined as the largest of R, G, or B. It runs from 0% to 100%.

If we plot the gamut of this system in three-dimensional space, do we get a "bicone"? No, we get just a cylinder.

What about the limitation in the range of saturation as we approach maximum luminance?

Well, S is not saturation, and more importantly, B is not luminance. But in any case, all possible values of H, from 0° to 360°; all possible values of B, from 0% to 100%; and all possible values of S, from 0% to 100% can coexist in any possible combination within an RGB gamut.

Thus a plot of the range of this "color space" forms a cylinder, with height 100% and radius 100%.

The bottom line

So, how should we feel about "bicone" representations? Just as we would about biographies of leprechauns.

The "bi-hexcone"

By the way, there is another fanciful solid figure often mentioned in this regard, the "bi-hexcone". It comprises two hexagonal pyramids, rather than circular cones, joined at their bases. It emerges from a different coordinate used for the radius, based on a different outlook on "saturation" (but of course, as before, not really).

How should we feel about hexcones? Just as we would about biographies of hexagonal leprechauns.

Best regards,

Doug
 
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