It has long been known that the visually perceived positions of objects, in short visual space, are distorted with respect to the physical positions. On the basis of the observation that equidistance-alleys lie outside parallel-alleys, Luneburg (Mathematical Analysis of Binocular Vision, Princeton University Press, Princeton, NJ, 1947) proposed that visual space is a Riemannian space of constant negative curvature. Luneburg used this observation, along with some additional assumptions, as an axiom and deduced the metric of visual space theoretically. Many researchers have tried to verify Luneburg's model experimentally, but the results are ambiguous. Moreover, many assumptions that Luneburg made were proved wrong in the literature. In this paper we will derive metric models for both visual and haptic space based directly on visual and haptic experiments involving a parallelity task. From the metric we make predictions for the egocentric distance, frontoparallel horopters, parallel- and equidistance-alleys and compare them to the literature. We also compare the metric structures of haptic and visual space. Both models provide a good description of the data from the parallelity tasks, even though the so-called oblique effect observed experimentally has not yet been incorporated.