A local fundamental theorem for projections
A projection is a mapping between linear spaces, which preserves collinearity, and whose restriction to any line is injective or constant. Motivated by applications in photogrammetry, the author proves that any projection from a subset \(M\) of a (little) Desarguesian projective space to another such space is a product of a central projection and an isomorphism, provided \(M\) is not too small and the image of \(M\) is not contained in one line. This leads to a new, extended proof of the following theorem (the case \(\dim P<\infty\) of which has been proved earlier by \textit{H. Lenz} [Math. Nachr. 18, 346-359 (1958; Zbl 0084.180)]): Let be given two real projective spaces \(P\), \(P'\), an open subset \(M\subset P\) and a projection \(\varphi:M\to P'\), the image \(\varphi(M)\) of which contains at least three non-collinear points; then \(\varphi\) is described by a linear mapping between homogeneous coordinates of \(P\), \(P'\).
scientific article; zbMATH DE number 147287