Duality, rigidity and planar parallax

We investigate the geometry of two views of seven points, four of which are coplanar, and the geometry of three views of six points, four of which are coplanar. We prove that the two are dual, and that the fundamental geometric constraints in each case are encapsulated by a planar homology. The work unifies a number of previously diverse results related to planar parallax, duality and planar homologies. In addition, we make a number of practical contributions, including for- mulae for computing the distance of the cameras from a distinguished world plane and formulae for structure computations. We show that the trifocal tensor is obtained uniquely from three views of six points, four of which are coplanar, and give a simple interpretation of the trifocal geometry. We give examples of these computations on real images.