Editing HSL and HSV (section)

===== HSL to RGB alternative =====

The polygonal piecewise functions can be somewhat simplified by clever use of minimum and maximum values as well as the remainder operation.

Given a color with hue <math>H \in [0^\circ,360^\circ]</math>, saturation <math>S=S_L \in [0,1]</math>, and lightness <math>L \in [0,1]</math>, we first define the function:
: <math>f(n) = L - a \max(-1, \min(k-3,9-k,1))</math>

where <math>k,n \in \mathbb R_{\geq 0}</math> and:

: <math>k = \left(n+\frac{H}{30^\circ}\right) \bmod 12</math>
: <math>a = S_L \min(L,1-L)</math>

And output R,G,B values (from <math>[0,1]^3</math>) are:

: <math>(R,G,B) = (f(0), f(8), f(4))</math>

The above alternative formulas allow for shorter implementations. In the above formulas the <math>a\bmod b</math> operation also returns the fractional part of the module e.g. <math> 7.4 \bmod 6 = 1.4</math>, and <math>k \in [0,12)</math>. 

The base shape <math>T(k) = t(n,H) = \max(\min(k-3,9-k,1), -1)</math> is constructed as follows: <math>t_1 = \min(k-3,9-k)</math> is a "triangle" for which values greater or equal to −1 start from k=2 and end at k=10, and the highest point is at k=6. Then by <math>t_2 = \min(t_1,1) = \min(k-3,9-k,1)</math> we change values bigger than 1 to equal 1. Then by  <math>t = \max(t_2,-1)</math>  we change values less than −1 to equal −1. At this point, we get something similar to the red shape from fig. 24 after a vertical flip (where the maximum is 1 and the minimum is −1). The R,G,B functions of {{mvar|H}} use this shape transformed in the following way: modulo-shifted on {{mvar|X}} (by {{mvar|n}}) (differently for R,G,B) scaled on {{mvar|Y}} (by <math>-a</math>) and shifted on {{mvar|Y}} (by {{mvar|L}}). 

We observe the following shape properties (Fig. 24 can help to get an intuition about them):

: <math>t(n,H) = -t(n+6,H)</math>
: <math>\min\ (t(n,H), t(n+4,H), t(n+8,H)) = -1</math>
: <math>\max\ (t(n,H), t(n+4,H), t(n+8,H)) = +1</math>