Estimate of the error committed in determining the place from where a photograph has been taken. (English translation of the former paper)
According to the Chasles theorem, the geometric locus of the points X for which the cross ratio of the lines XA, XB, XC and XD is constant (for four given points A, B, C and D):
r = (XA, XB, XC, XD)
is a conic passing through the points A, B, C and
The Chasles theorem applied to a photograph allows to calculate the point
where it was taken in the following way. We must identify five buildings
in the photograph with a known location on a chart. Then, we calculate the
cross ratios of two sets of four verticals of the buildings in the photograph.
The geometric locus of the points X and X' with these cross
ratios on the map are two conics whose intersection is the searched point.
In the paper, we study the propagation of the initial measure errors to the calculated coordinates of the point of view of the photograph. The differential technique is applied to an expression of the cross ratio obtained by means of the geometric algebra:
r = (XA^XC) (XB^XD) / (XA^XD) (XB^XC)
where ^ is Grassmann's exterior product of the indicated vectors.