how to understand the singularity of saturation by RGB(0,0,0)

Question

i have not found an article to explain why by transofrmation RGB Color model to HSI we have undefinde Saturation when the color is black? Qeustion1: what is the explation of this (the mathematic reason)

Question2: when the singularity is in the black color, that means we could not define what is the Saturation. but the Question then why the HSI is sensitiv to error also when the Saturation is small(not Zero but in the near from Zero). i have read that it is better not allow the saturation to be very small?

Answer 1

The mathematical reason is an indeterminate form 0/0. It should be intuitive that "there is nothing to see" in black.

Answer 2

The saturation of RGB 0,0,0 cannot be calculated and is therefor defined as zero.

S = (max(r,g,b) - min(r,g,b)) / max(r,g,b)

You see that r=g=b=0 would result in a problem as we cannot devide by 0.

The formula also shows that very small saturation values can only occur if we have very similar RGB values. If a pixel has a low saturation it is "more gray" or more achromatic. It doesn't make much sense to apply colour based rules to non-colours.