Previous empirical studies have shown that information along visual contours is known to be concentrated in regions of high magnitude of curvature, and, for closed contours, segments of negative curvature (i.e., concave segments) carry greater perceptual relevance than corresponding regions of positive curvature (i.e., convex segments). Lately, Feldman and Singh (2005, "Psychological Review, 112", 243-252) proposed a mathematical derivation to yield information content as a function of curvature along a contour. Here, we highlight several fundamental errors in their derivation and in its associated implementation, which are problematic in both mathematical and psychological senses. Instead, we propose an alternative mathematical formulation for information measure of contour curvature that addresses these issues. Additionally, unlike in previous work, we extend this approach to 3-dimensional (3D) shape by providing a formal measure of information content for surface curvature and outline a modified version of the minima rule relating to part segmentation using curvature in 3D shape. (Contains 25 footnotes and 8 figures.)