Since each retinal projection of an object has some definite geometric relation to it, the two projections of a same object are related to each other. The correspondences between points, linear segments, and orientations on the two retinas are discussed thoroughly in three-dimensional geometry, and the constraints that tie the projections (the 'projective invariants') are presented. Some widely held conceptions on disparity or the ambiguity problem in binocular stereopsis appear to be based upon representation that are correct in particular situations, but are misleading in the general case. I suggest that in order to achieve binocular stereopsis, the brain does not proceed by complete trial and error, but may guide its search for correspondences by taking advantage of the geometric constraints. There are at least four major possible strategies: (i) a metric strategy, as initially proposed by Julesz; (ii) a projective strategy based on the law of invariance of the anharmonic ratio and Desargue's theorem; (iii) a perspective strategy discussed in relation to the homology relationships between vanishing points and in relation to physiological studies on cells of visual cortex; and (iv) a more dynamic strategy based upon the geometric properties of the Zöllner illusion.