1 Introduction In daily life, we identify the visually perceived world with the physical world. The reason for this is straightforward: visually directed behaviour, such as knowing whether an object is in grasping range, does not pose any serious problems. Scientifically, this identification is not so trivial unless the perceived spatial relations, such as the perceived distance between objects, the perceived straightness of a line, and so on, resemble the physical spatial relations. However, throughout the literature it has been shown that the perceived geometrical relations deviate significantly from the physical relations. For example, Helmholtz (1867/1962) showed that wires which are arranged in an apparent frontoparallel plane do not lie in a physically frontoparallel plane; Hillebrand (1902) found that two lines in depth which appear equidistant are not physically equidistant; and Blumenfeld (1913) found that such equidistant lines do not appear straight and parallel. Evidently, the visually perceived space is distorted with respect to physical space. The correspondence is probably systematic, for otherwise it is hard to understand why visual perception does not pose problems in daily life. However, it is doubtful whether there is a one-to-one correspondence. Therefore, we still have to determine which geometrical relations exist between visual percepts and find out to what extent visual space corresponds to physical space. In Euclidean geometry of the plane, the sum of the interior angles of a triangle equals p radians and parallel lines are equidistant. It was found, however, that these relations do not hold in visual space (Blumenfeld 1913; Foley 1980, 1991; Indow 1991) and that visual space therefore cannot be described by a Euclidean geometry (Luneburg 1947; Large systematic deviations in visual parallelism Perception, 2000, volume 29, pages 1467 ^ 1482 Raymond H Cuijpers, Astrid M L Kappers, Jan J Koenderink Helmholtz Instituut, Universiteit Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands; e-mail: R.H.Cuijpers@phys.uu.nl , A.M.L.Kappers@phys.uu.nl Received 8 February 2000, in revised form 3 October 2000 Abstract. The visual environment is distorted with respect to the physical environment. Luneburg [1947, Mathematical Analysis of Binocular Vision (Princeton, NJ: Princeton University Press)] assumed that visual space could be described by a Riemannian space of constant curvature. Such a space is described by a metric which defines the distance between any two points. It is uncer- tain, however, whether such a metric description is valid. Two experiments are reported in which subjects were asked to set two bars parallel to each other in a horizontal plane. The backdrop consisted of wrinkled black plastic sheeting, and the floor and ceiling were hidden by means of a horizontal aperture restricting the visual field of the subject vertically to 10 deg. We found that large deviations (of up to 408 ) occur and that the deviations are proportional to the separa- tion angle: on average, the proportion is 30%. These deviations occur for 308, 608, 1208, and 1508 reference orientations, but not for 08 and 908 reference orientations; there the deviation is approximately 08 for most subjects. A Riemannian space of constant curvature, therefore, cannot be an adequate description. If it were, then the deviation between the orientation of the test and the reference bar would be independent of the reference orientation. Furthermore, we found that the results are independent of the distance of the bars from the subject, which suggests either that visual space has a zero mean curvature, or that the parallelity task is essentially a monocular task. The fact that the deviations vanish for a 08 and 908 orientation is reminiscent of the oblique effect reported in the literature. However, the `oblique effect' reported here takes place in a horizontal plane at eye height, not in a frontoparallel plane. DOI:10.1068/p3041