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(401) 
THE “CRITICAL SURFACES” IN THE PHOTOGRAMMETRIC 
CARDINAL PROBLEM AND THEIR IMPORTANCE AND 
ELIMINATION IN PHOTOGRAMMETRIC PRACTICE 
by 
Dr. W. Hofmann. 
This report is a summary of the paper on the “Problem of Critical Surfaces 
in Theory and Practice“ which will be published this year by the German 
Geodetic Commission. It contains only an exposition of the most important 
results and omits their rigid development by mathematical calculation. 
After previous investigations by E. C. P. Poivilliers, R. Bosshardt and 
R. Finsterwalder, the first general solution to the problem of “critical surfaces” 
has been supplied by J. Krames. He found by projective-geometric research 
chat the relative orientation of a picture pair becomes uncertain if 
1) the photographed terrain has the form of an orthogonal regular surface of 
the second order (orthogonal hyperboloid, paraboloid, cone, rotating cy- 
linder, hyperbolic cylinder, pair of planes), and if 
2) the photographic base lies on a so called “principal determinant” of the 
surface; it may be assumed that an orthogonal-regular surface is produced 
by two coincident congruent bundles of planes supported by the principal 
determinant. 
A discussion as to which of the 8 possible orthogonal regular surfaces of 
the second order may actually produce a critical case in aero-photogrammetric 
practice, must start from photographic and morphological possibilities. Critical 
formations of the terrain may occur in valleys. A valley formation of this type 
must have an adequate length and width so as it can fully cover one pair of 
aerial photographs. Furthermore, the valley must not open into side valleys 
within the area, covered by a picture pair. 
Otherwise, control points outside the critical area will be available for 
unequivocal relative orientation. Since a photographic flight is usually made on 
a level parallel to the bottom of the valley and at a more or less constant height 
above the bottom, the orthogonal hyperboloid and the orthogonal cone are 
eliminated right away as critical surfaces for all practical purposes. In either 
case, the principal determinants are in an oblique position with reference to 
the valley bottom as regards position and elevation. They may therefore not 
be used as flight paths or photographic bases. 
The principal determinants of the orthogonal paraboloid, which must be 
on the ground itself, cannot be bases for an air photograph. This also applies to 
the hyperbolic cylinder and the orthogonal pair of planes. Hence, there remains 
only the rotating cylinder as a critical surface, provided the base of the photo- 
graph is on one of the straight lines of its convex surface. The rotating cylinder 
on which the base is on the vertex convex surface line has already been recog- 
nized by Bosshardt and Finsterwalder to be the most important critical surface. 
In practice, the hyperboloid and the cone may become critical surfaces if their