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ANALYTICAL AERIAL TRIANGULATION, SCHUT 13 
From this we may conclude that in a sound theory of analytical aerial triangulation 
the three conditions employed by the above four practical methods lead to only one con- 
dition equation: 
BXX,-X 
i377 0, 
which is the correct analytical formulation for the condition that corresponding rays 
must intersect. 
5. Three orientation elements are needed to determine the attitude of each photo- 
graph and along with it the components of the vectors X, and Xii on the axes of an 
orthogonal strip coordinate system. 
All four of the above methods use these three elements as three independent para- 
meters for the construction of an orthogonal transformation matrix. In terms of vector 
analysis the elements of this matrix are the orthogonal components of a dyadic. The 
vector X, with components X,, Y, and Z, in the strip coordinate system follows from 
the transformation 
X; AQ Xp (k= 1 i+1) 
in which A, is this dyadie and the vector x, has as components the photograph coor- 
dinates, with respect to an origin in the principal point, and the focal length. 
Several methods by which the orthogonal matrix can be constructed from three 
independent parameters have been listed in reference [2]. The most suitable selections 
of three parameters are: 
a. three rotations a,, a, and a; about orthogonal axes through the projection centre of 
the photograph and parallel to the X-, Y- and Z-axis respectively or the sines of these 
rotations, 
three independent elements of the orthogonal matrix, 
C. À sin a, u sin a and v sin a, in which a is a rotation about a directed line and 2, u and 
» are the direction cosines of this line with respect to the X-, Y- and Z-axis respec- 
tively, and 
d. the components a, b and c of either the unit quaternion 
d t ai- bj +ek (d? 4 a? + b2 + e2 = 1) or of the quaternion 
1 ai + bj +ck. 
In the Ordnance Survey Method three elements of the orthogonal matrix are selected 
as parameters. In the Schmid and the Herget Methods the three rotations o,, a» and ag 
are selected, and in the author's method the sines of these rotations. The use of quater- 
nion parameters may be a trifle more economical. 
On solving the condition equations. 
6. Whichever the choice of orientation elements, the condition equations are non-linear 
in these elements. In order to compute the orientation elements the four methods replace 
the condition equations by linear equations and solve these. 
In a sound method the linear equations express the condition equations as well as 
possible. Geometrically speaking each condition equation is represented by a curved 
hypersurface in a space which has one coordinate axis for each of the orientation ele- 
ments. The best linear approximations of the hypersurfaces are hyperplanes tangent to 
the hypersurfaces in the area of the solution. These hyperplanes are the geometrical 
representation of linear equations derived from the condition equations by differentiation 
with respect to the orientation elements and substitution of approximate values in the 
coefficients. Therefore the linearization must be achieved by differentiation or by any 
procedure which gives the same result. 
If linearization should give a different result the representative hyperplanes inter- 
sect the curved surfaces instead of being tangent to them. Their intersection can be