This paper demonstrates the existence of a new,
approximate, intrinsic ambiguity in Euclidean structure from motion (SFM)
which occurs as generically as the bas--relief ambiguity but, unlike it,
strengthens for scenes with more depth variation. The ambiguity is absent in
projective SFM, but at the cost of an increased likelihood that the
projective reconstruction has very large errors. The analysis gives a
semi--quantitative characterization of the least--squares error surface over
a domain complementary to that analyzed by Jepson, Heeger, and Maybank. As
part of our analysis, we show that the least--squares error for
infinitesimal motion---the optical--flow error---gives a good approximation
to the least--squares error for moderate finite motions. We also demonstrate
the existence of a new local minimum in minimizing  over the rotation
given the translation direction.