It is proposed that any cue for the visual discrimination of shape, in particular the discrimination of curved contours, should be such that the perceptual relationships defined by the cue are invariant under changes in observer viewpoint. Such relationships may be quantified by the Weber fraction; that is, the ratio Δc/c, where, for any particular value c of the cue, Δc is the smallest difference in c that can be detected. Eight geometric attributes of curved contours having one symmetry axis and parallel chords (a standard stimulus configuration) were examined for invariance of the Weber fraction under symmetry-preserving affine transformations of the image plane (changes in viewpoint are well approximated by affine transformations when depth is small relative to viewing distance). The attributes, each investigated in previous psychophysical studies, were equivalent-curvature, radius-of-curvature, turning-angle, arc-length-divided-by-chord-length, arc-length, maximum-deviation (sag), area, and mean-deviation. Three of the attributes, namely sag, area, and mean-deviation, satisfied the viewpoint-invariance condition; the remainder failed. These results are considered in relation to previously published empirical data on the Weber fraction for contour-curvature discrimination. © 1993 Springer-Verlag.