The HSI color model is a very influential theoretical model in color model theory and is widely used in the research of hue-preserving color image processing. However, it is common that the introduction of this model focuses on theory and concepts, while ignoring some of its disadvantages, and even has some vague or wrong explanations. The paper explains the geometric space of the HSI color model and the geometric meaning of its three components Hue, Saturation, and Intensity from the perspective of analytic geometry, to correct some common misunderstandings about the HSI model. For the inevitable gamut problem when processing color images in the HSI model, the paper proposes an effective compression method to smoothly compress the colors beyond the RGB color cube back to the color cube while keeping the majority of other colors unaffected by the compression, thereby protecting the effect of the processing transformation.