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What is the basic practical difference between CIE Lab and CIE Lch (aka HCL, right?) color models?

All I have found about them on Google is that they are two different ways to represent the same color space. But there are no explanations about why the two different ways are necessary at all, and there are also no examples to see where one model is better than another.

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This direct screenshot from Wikipedia shows the conversion between Lab and HCL.

enter image description here

HCL is only for convenience, it doesn't change the color space (except in case the numbers are limited to sparse integer values)

What convenience?

For ex. I like to use HCL version because I can imagine the color changes more easily beforehand by adjusting hue and chroma of HCL. Trying to guess how to change a and b in Photoshop's Lab system for a wanted change is much more trial and error than using HCL which is available in GIMP.

Fortunately in Photoshop I can also adjust hue and chroma. The hue/saturation adjustment works with hue and chroma in Lab mode, but the colors are in the color picker shown as Lab-numbers.

I prefer HCL over Lab if I can choose because the bones of my head are too thick for intuitive handling of Lab-numbers. But there exists people who have things differently. See for ex this old case:

Is there a way to remove lens flare?

Hopefully you have already noticed that Lab and HCL are made to present the colors which humans can see so that the luminance, chroma and hue are visually separated from each other. Actually in Lab hue and chroma are mangled together in a and b, but L is independent.

RGB and its polar variations (which are made also for covenience) present commands for colored light production machinery in computer screens and what RGB sensors in a camera output. The visual effect of the saturation in polar RGB systems like Photoshop's HSB depends drastically on hue and brightness. In HCL the factors are much more visually independent. That has an unfortunate consequence: It's far too easy to select in Lab or HCL a color which cannot be produced in RGB screens. Fortunately the programs have options to switch out of gamut warning ON.

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    One could also add that HCL sort of is to Lab what HSB/HSV is to RGB.
    – Wolff
    Dec 6 '21 at 17:19
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    Both of them may be available when you tweak colors in programs, but someone can tell to you a color in Lab or you must tell the color in Lab. That's nothing neverbeforeheard, because Lab tells something of the visual look of a colored material, not what a computer does. HCL, of course does the same, but I have never seen the colors of materials told in HCL
    – user287001
    Dec 6 '21 at 18:03
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    @johnc.j.(continued) Just a warning for the future: There's no valid general method to reliable calculate the Lab color of a mixture of paints or color powders which all have their colors specified in Lab system. I's asked here now and then, but it's useless. Every material behaves differently.
    – user287001
    Dec 6 '21 at 18:13
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    Nope. You should do your best to understand both of them and perform tasks with both of them. Nobody knows when you suddenly cannot select, but still must get the job done. Besides it can well happen that you finally have no difficulties to adjust Lab numbers intuitively (see the linked story of lens flare removal in my answer) .
    – user287001
    Dec 6 '21 at 21:15
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    @johnc.j. the axes of Lab coincide with how the adversarial nature of human senses work. By working in Lab its easier to boost certain aspects and deal with colorblindness issues. You can easily do this with curves in Lab but not curves in LCH. On the otherhand Lch is better at picking colors for certain uses.
    – joojaa
    Dec 6 '21 at 21:57

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