The critical line congruence for reconstruction from three images

Let three projected images of a set of lines in space be given. Then, in general, the relative positions of the lines can be reconstructed uniquely up to a collineation of space. Reconstruction fails to be unique for certain critical sets of lines. It is known that each critical set is parameterised by a Bordiga surface in ℙ4. A new proof of this result is given. In addition, it is shown that every Bordiga surface parameterises a critical set of lines. The proof involves an explicit construction of the second or spurious set of lines which projects down to the same three images as the veridical set of lines.

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