In this question it was asked whether it was possible to reproduce any color-only change to an image using just curves. In my answer, I said that there were color-only changes such as conversion to monochrome, changes to saturation, and others that couldn't be reproduced using curves as they involve more than just adjusting the input-to-output curve of the 3 separate channels, or even all 3 at once.
RyanFromGDSE commented that my point was incorrect. So, I'd like to understand how one can use just curves for more complex color changes. Here are some examples that don't seem immediately obvious to me:
To convert to monochrome, I've used the Channel Mixer adjustment in Photoshop and for each output channel set it to be 21% of the input red, 72% green, and 7% blue (so roughly the same as the conversion for Rec. 709 Luma).
This transformation involved not only scaling each of the channels but adding them together, and then distributing that result to all 3 channels. How does one achieve that with curves?
I'm also curious about other transformations like saturation. If I want to increase the saturation of blue without increasing the saturation of cyan and magenta, I don't see how to do that with curves. For example, if I have = <0.25, 0.25, .05> and I increase the blue, it becomes <0.25, 0.25, 0.75> and increases the saturation of that color. But if I started with <0.6, 0.25, 0.5>, after applying the curve, I'd end up with <0.6, 0.25, 0.75> which is not a more saturated version of the original color. It's just a more blue color.
I'm also curious about others, such as channel swapping, or something like a gradient colorize where the image is converted to monochrome and then a 1D LUT is used to replace gray levels with a set of colors in the table.
So it's become clear that the case originally referred to meant using curves in a hue/saturation (or if you prefer luma/chroma) color space, such as L a* b*. Since that involves converting to a different color space and then applying curves, it is not using just curves, so doesn't satisfy my original assertion.