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When Photoshop performs a "Profile-to-Profile" conversion between the HOT and sRGB color spaces, it successfully translates the digital codes for all of the colors that each system has in common. What about the colors in HOT that can't be represented in sRGB (they would result in negative or overflow values)? The colors involved are considerable: the area inside the HOT triangle (the HOT color gamut) but outside the sRGB triangle in the chromaticity diagram (the sRGB gamut) are the colors we are talking about. This is where gamut-mapping comes in. We need to find suitable substitutes to use to represent those colors, because they can't be made on the sRGB display. This is an active topic of current color research, but we know that the best substitute color depends on the purpose of the image. Different color substitution strategies will be perceived as better, depending on what the image is for. This is the origin of the choice of "rendering intents" embodied within ICC profiles. Different color management "engines" (color matching method, CMM) will also provide different ways of handling the gamut mapping problem. This is one reason there is a market for different CMMs, different profiles, and different profile-making tools. There is no single best answer, each application must find its own satisfactory solution. In my case, I wanted to see the hues associated with the HOT primaries. I was interested in a strict hue-preserving gamut mapping: a substitute color can have a different lightness or saturation, but the hue must stay correct. I also wanted "soft clipping". As colors become out-of-gamut, they should be gradually remapped, not suddenly clipped to some constant substitute. The purpose for this is obvious: I am interested in images, where colors tend to shade into each other and naturally fall off toward shadows and highlights. To achieve "soft-clipped, constant-hue" gamut mapping, I had to write my own profile-to-profile mapping program that incorporated these rules. Here are some examples of the difference between the Photoshop gamut mapping process, and the custom processing that strictly preserved hue.
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