different kind of saturation

there are many different ways of tuning saturation, darktable does offer a few alternative ways to alter saturation and the reason for this post is to clarify what they do and how they work. the image on left is the original untouched image used for the different examples below, use it as a reference for comparing the results of the different kind of saturation described below, the resulting effects is exaggerated to make it easier to spot the differences.