Figure 6 presents the object-color solid thus obtained. The solid black and gray contours in
Figure 6 correspond to constant values of β and γ, respectively. Note that this regular sampling of β and γ yields a highly irregular grid on the object-color-solid boundary. In other words, a regularly spaced collection of directions (rays) in
k-space brings about rather irregularly spaced beam of vectors in
SML space. This irregularity is easily seen in
Figure 7A, where the resulting sample of
SML vectors is presented as the points in the chromaticity plane (same as in
Figure 2) corresponding to these vectors. Although the cluster of the chromaticity points in this graph results from a homogeneous sample of the polar angles (qualitatively similar results are found when using geodesic sampling), there are vast areas in the chromaticity plane void of markers despite a rather dense
k-sampling resulting in 1,288,417
ks. This alone is enough to undermine the use of
k-sampling as a practical way of sketching the object-color solid; however, that is not the only reason. As can be seen from
Figure 4, many of the
k-samples generate either zero- or one-transition reflectances. Specifically, for this sample of polar angles defining a set of
k-samples the zero-, one-, two-, three-, and four-transition reflectances arise from 27.2%, 55.1%, 17.2%, 0.42%, and 0.008% of the
k-samples, respectively. Since the chromaticity loci corresponding to the zero- and one-transition reflectances comprise the poles and a curve, that is, occupy a zero-area fragment of the chromaticity diagram, only 18% of the
k-samples are actually mapped onto the vast majority of the chromaticity gamut. Given the extremely nonlinear relationship between the rays in
k-space and vectors in
SML space, we must conclude that
k-sampling is highly ineffective when using it for evaluating the object-color-solid boundary.