Critical configurations for 1-view in projections from P-k -> P-2

In this paper we describe, from a theoretical point of view, critical configurations for the projective reconstruction of a set of points, for a single view, i.e. for calibration of a camera, in the case of projections from P-k to P-2 for k >= 4. We give first a general result describing these critical loci in P-k, which, if irreducible, are algebraic varieties of dimension k-2 and degree 3. If k=4 they can be either a smooth ruled surface or a cone and if k = 5 they can be a smooth three dimensional variety, ruled in planes, or a cone. If k >= 6, the variety is always a cone, the vertex of which has dimension at least k - 6. The reducible cases are studied in Appendix A. These results are then applied to determine explicitly the critical loci for the projections from P-k which arise from the dynamic scenes in P-3 considered in [13].