CIEHLC Cylindrical Colour Spaces

The CIELCh or CIEHLC is a colour space based on either CIELUV or CIELAB, which uses polar coordinates $C^{\ast }$ (chroma/relative saturation) and $h°$ (hue angle/angle of hue in the CIELUV/CIELAB colour wheel) instead of the cartesian coordinates $u^{\ast }$ and $v^{\ast }$/$a^{\ast }$ and $b^{\ast }$. The brightness $L^{\ast }$ remains unchanged.

${\text{CIELCh}}_{ab}$ - Cylindrical CIELAB ${\text{CIELCh}}_{uv}$ - Cylindrical CIELUV

Conversion

For CIELUV HLC $C_{uv}^{\ast }=\sqrt{{u^{\ast }}^2+{v^{\ast }}^2}$
$h{°}_{uv}=atan(v^{\ast }/u^{\ast })$

Furthermore, saturation correlate can be written as: $s_{uv}=\frac{C_{uv}^{\ast }}{L^{\ast }}=13\sqrt{(u^{\prime }-u_n^{\prime })+(v^{\prime }-v_n^{\prime }{)}^2}$
No such correlation for saturation exists for CIELAB HLC

For CIELAB HLC: $C_{ab}^{\ast }=\sqrt{{a^{\ast }}^2+{b^{\ast }}^2}$
$h{°}_{ab}=atan(b^{\ast }/a^{\ast })$

$a^{\ast }=C^{\ast }cos(h°)$ $b^{\ast }=C^{\ast }sin(h°)$