Surface shape and curvature scales

摘要

The classical surface curvature measures, such as the Gaussian and the mean curvature at a point of a surface, are not very indicative of local shape. The two principal curvatures (taken as a pair) are more informative, but one would prefer a single shape indicator rather than a pair of numbers. Moreover, the shape indicator should preferably be independent of the size i.e. the amount of curvature, as distinct from the type of curvature. We propose two novel measures of local shape, the ‘curvedness’ and the ‘shape index’. The curvedness is a positive number that specifies the amount of curvature, whereas the shape index is a number in the range [−1, +1] and is scale invariant. The shape index captures the intuitive notion of ‘local shape’ particularly well. The shape index can be mapped upon an intuitively natural colour scale. Two complementary shapes (like stamp and mould) map to complementary hues. The symmetrical saddle (which is very special because it is self-complementary) maps to white. When a surface is tinted according to this colour scheme, this induces an immediate perceptual segmentation of convex, concave, and hyperbolic areas. We propose it as a useful tool in graphics representation of 3D shape.