Given , and , we can first find , , and on the RGB triangle, and the find , , and . Consider three possible cases in terms of the hue angle :

- ( and , i.e., inside
)
From , solving for we get

To find , consider the triangle :

Here vector is the projection of vector onto the line , and all points , , and are on the RGB triangle. Now consider the following steps:

- On the triangle , we have

and

(* and ) - The length of is
, and the saturation
is , i.e., , we have:

and we get

- As
, i.e.,
, and also
, we have

Substituting this into the expression of above, we get:

- Finally we can get

Given and , we can get :

- On the triangle , we have
- (p inside
)

- (p inside
)

In summary, given the H, S, and I values, we can obtain the R, G, and B
components by

and

Note that after the HSI to RGB conversion, we need to convert the R, G, and B values from the range of back to the three components for a pixel in a color image in the range of .