Communication to the 
IXth INTERNATIONAL CONGRESS OF PHOTOGRAMME TR Y 
COMMISSION II 
Reprint from 
The Photogrammetric Record, Vol.III, No. 15, 1960 
DETERMINATION OF THE FLATNESS OF A 
SURFACE IN COMPARISON WITH A CONTROL 
PLANE 
By B. llALIFRT 
Summary 
The problem is to determine the flatness (the deflections) of the surfaces 
in comparison with a control plane of known and high precision, for instance 
a surface plate. The lack of flatness of the surface can be demonstrated with 
respect to a plane through three points which must be made accurately parallel 
to the control plane (the plate). The accuracy of the determination of the 
deflections of the surface is demonstrated and a criterion is constructed for 
judging whether an arbitrary point of the surface deviates significantly from 
the reference plane or not.1 
1. THE MEASUREMENTS 
The measurements of the elevations of the points of the surface (Fig. 1) are 
assumed to be performed with a measuring gauge, which is supported by the control 
plane. The possible systematic errors of the measuring gauge must be accurately 
determined so that the readings can be corrected. The surface to be measured is 
adjusted in a position which is approximately parallel to the control plane. This 
is checked with measurements in the points 1-3. 
The elevations of the points 1-3 and all other wanted points of the surface 
are then measured. The measurements should be made with the same accuracy 
in all points, for instance by repeating the settings a number of times (e.g. three 
times) and then computing the average for each point. From the deviations v; — v, 
of the individual readings from the average the standard deviation of each individual, 
measurement and of the average can be computed according to well-known formulae. 
t Clearly, lack of flatness of images is of particular interest for photogrammetry, since the image co-ordinates 
will become influenced due to the central projection. As a matter of fact there must be a certain weight distribution 
within the image due to the inevitable deflections and to the errors of the measurements of the flatness.