Visual globes, celestial spheres, and the perception of straight and parallel lines

Helmholtz's famous distorted chessboard pattern has been used to make the point that perception of the straightness of peripherally viewed lines is not always veridical. Helmholtz showed that the curved lines of his chessboard pattern appear to be straight when viewed from a critical distance and he argued that, at this distance, the contours stimulated particular 'direction circles' in the field of fixation. We measured the magnitude of the distortion of peripherally viewed contours, and found that the straightness of elongated contours is indeed misperceived in the direction reported by Helmholtz, but that the magnitude of the effect varies with viewing conditions. On the basis of theoretical considerations, we conclude that there cannot, in principle, be particular retinal loci ('loci' is used here in the sense of an arc or an extended set of points that provide a basis for judging collinearity) to underpin our judgments of the straightness and parallelity of peripheral contours, because such judgments also require information about the 3-D surface upon which the contours are located. Moreover, we show experimentally that the contours in the real world that are judged to be straight and parallel can stimulate quite different retinal loci, depending on the shape of the 3-D surface upon which they are drawn.