The image-plane \(Y\) of a central axonometry is embedded into an oriented real Euclidean 3-space \(X\) which is extended projectively. The {central-axonometric reference system} consists of the image \(o\in Y\) of the origin, the images \(e_1,e_2,e_3\in Y\) of the unit points, and of the vanishing points \(f_1,f_2,f_3\in Y\) of the axes such that:
1. the 7 points \(o,e_1,e_2,e_3,f_1,f_2,f_3\) are non-collinear,
2. \(o,e_1,e_2,e_3\) are affine points, and 3. each triple \((o,e_j,f_j)\), \(j=1,2,3\) consists of pairwise distinct collinear points.
The author analytically proves a necessary and sufficient condition for \(o,e_1,e_2,e_3,f_1,f_2,f_3\) to be the image of a positively oriented orthonormal frame under a central projection of \(X\); the condition is a single equation which contains 3 complex numbers describing the mutual situation of \(o,e_1,e_2,e_3\) and the 3 cross-ratios CR\((o,e_j,f_j,u_j)\) with \(u_j\) denoting the point at infinity of the line \(o\vee\,e_j\).
From his equation the author derives the conditions from J. Szabó, H. Stachel and H. Vogel [Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 203, 3–11 (1994; Zbl 0841.51013)] where the vanishing points \(f_1,f_2,f_3\) are affine, and the condition from E. Stiefel [Lehrbuch der darstellenden Geometrie, 3rd ed. (1971)] where \(f_1,f_2\) are affine and \(f_3\) is a point at infinity; in contrast to the cited 2 works, the author needs no assumption concerning the type of \(f_1,f_2,f_3\). The author’s equation is an analogue of Gauss’ fundamental theorem of axonometry [cf. E. Stiefel, Lehrbuch der darstellenden Geometrie, 1st ed. (1947; Zbl 0031.28001), p. 50] (reviewer’s hint: compare also M. Eastwood and R. Penrose [Math. Intell. 22, No. 4, 8–13 (2000; Zbl 1052.51505)]).