CIELUV Ch (CIE LCh) or Polar Luv is cylindrical representation of CIELUV (and for CIELAB : Polar Lab)
HCL is implementation of CIELUV Ch
Problem of HCL: HCL tends to specify colors outside of the RGB gamut, which means that some sensible choices of HCL values will generate values outside the (0,0,0)-(1,1,1) RGB cube. More worryingly, the HCL->RGB transformation is discontinuous. Although it lies outside the RGB gamut, clamping the transform to the closest RGB point does not patch the seam in the color space. Example of Clamping
HCL to RGB need conversion chain : HCL->LUV->XYZ->RGB
HSLuv, other implementation of CIELUV Ch which fully maps to RGB. HSLuv is just like LCH except that it stretches the saturated colors for each hue so that any saturation coordinate represents a valid color, unlike LCHs chroma coordinate. The downside is that the colors chroma (colorfulness) can change when you drag the hue slider. HSLuv, is based on CIELCHuv, the cylindrical transformation of CIELUV; the LCH in David Johnstone's colour picker is CIELCHab, the cylindrical transformation of CIELAB.