"Estimació de l'error comès en determinar el lloc des d'on s'ha pres una fotografia."
Ramon González, Josep Homs, Jordi Solsona (Butlletí de la Societat Catalana de Matemàtiques 12, núm 1 [1997] 51-71).

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 D. 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.