Isochromatic lines as extension of Helmholtz reciprocity principle for effect paints

Flake-based parameters were recently introduced as a physical concept to predict a series of measurement geometries producing similar reflection data for effect paints. We derive expressions to calculate these so-called isochromatic lines, connecting the two Helmholtz-reciprocal in-plane geometries with a series of out-of-plane geometries. Thus isochromatic lines can be regarded as an extension of the Helmholtz reciprocity principle, which is valid for effect paints. We experimentally studied seven effect paint samples with large angular color variation along the length of four isochromatic lines. A change in illumination angles by up to 75° while following isochromatic lines led to a standard deviation in color parameters of less than two units. When isochromatic lines were not followed, these colorimetric parameters varied by more than 10 units already by change in detection angle of 10°. Therefore the concept of isochromatic lines works well for effect paints.