# Convert HSV to RGB colors

HSV colors are composed of a triple of numbers: hue $\in [0, 360)$ (in degrees), saturation $\in [0, 1]$ and value or brightness $\in [0, 1]$. RGB colors instead are more well-known and are also composed of a triple of numbers all of them in the range $[0, 1]$ or in a byte $[0, 255]$.

I was trying to implement an algorithm that converts HSV colors to the corresponding RGB ones, and I managed to do it based on the following algorithm provided by Wikipedia. (I'm reporting the whole related text).

$C = V \times S$

\begin{align} H^\prime &= \frac{H}{60^\circ} \\ X &= C (1 - |H^\prime \;\bmod 2 - 1|) \end{align}

($R_1$, $G_1$, $B_1$) = \begin{cases} (0, 0, 0) &\mbox{if } H \mbox{ is undefined} \\ (C, X, 0) &\mbox{if } 0 \leq H^\prime < 1 \\ (X, C, 0) &\mbox{if } 1 \leq H^\prime < 2 \\ (0, C, X) &\mbox{if } 2 \leq H^\prime < 3 \\ (0, X, C) &\mbox{if } 3 \leq H^\prime < 4 \\ (X, 0, C) &\mbox{if } 4 \leq H^\prime < 5 \\ (C, 0, X) &\mbox{if } 5 \leq H^\prime < 6 \end{cases}

Finally, we can find $R$, $G$, and $B$ by adding the same amount to each component, to match value

\begin{align} &m = V - C \\ &(R, G, B) = (R_1 + m, G_1 + m, B_1 + m) \end{align}

where $H$ is the hue, $V$ is the value, $S$ is the saturation, $C$ is what Wikipedia calls the chroma and $R, G, B$ are correspondingly the values of red, green and blue.

Before wanting to find a way to convert HSV colors to RGB Colors, I had already a formula to convert RGB colors to HSV, which I wanted to manipulate to obtain a inverse formula, but I didn't manage to do it.

Here's a picture:

I could show you my attempts, but they are not useful.

## Question

• How is the formula (provided by Wikipedia) to convert a HSV to RGB colors derived from the formula in the picture above to convert RGB to HSV colors?

I'm asking this question here because I'm looking for a mathematical manipulation of one formula to arrive at another. I'm also asking this question because I think it could be interesting also for others. If you think this question should be moved to another stackexchange's website, feel free to migrate it or just tell me and I will post it where you indicate me.