14 ANALYTICAL AERIAL TRIANGULATION, SCHUT
farther from the intersection of the curved surfaces than the intersection of the tangent
planes. This will have an adverse effect upon the speed of convergence of the iteration
procedure. Such a linearization cannot be accepted as part of a sound method.
In the first and second triangulation procedures differentiation of the condition
equation B X X; X;,,— 0 with respect to the orientation elements leads to the linear
correction equation
OX, V OX, aV OX 4
BxX- (oa a, + BEX X, | TEl| aad BOX Xi € Y a3 +
A da, 2 À da 2 V O05
+ (B°+ dB) X X; X9,4 = 0
in which:
4, , 49 and ag are three independent parameters which determine an orthogonal
transformation matrix,
dB is the correction to the base vector, and
the superscript ? indicates approximate values of the elements, not unit vectors as in
Herget’s equation.
in which A*;, , and X*; , , follow from assumed approximate values of the parameters in
In the third triangulation procedure the condition equation must be differentiated
also with respect to the matrix parameters of photograph i, with obvious results.
The vectors X,;,, can be computed as
X17 AU XS with X*;,,— A*,,* Xi i1
the matrix of A, , ,. Thus a,, a, and a, are three independent parameters in the matrix
of the dyadic Ac which completes the orientation. They will be small quantities and
therefore in the above derivatives of > zero has been chosen as the approximate value
for each of them.
In the four constructions listed in paragraph 5, let the parameters à,, ad, and a,
represent
a. ay, as and ay, respectively, or their sines,
b. —453, + a5 and —ay,, respectively,
c. À sina, u sina and v sin a, respectively,
d. a, b and c, respectively,
and let i, j and k represent unit vectors in the direction of the X-, Y- and Z-axis,
respectively.
Then in each of these four cases, differentiation of A^ gives
à Ac S ; 0 Ac ; j
TET = —a,jk + a,ki, de = +agik — Aoki,
0 Ac + i:
So; = —Gsl] + ag,
and so:
0X,
i+ 1 = . 7
0a, T 441) s Y;,k
OX;+1 7 : X k
Ö de A
0X, | y DpSX 4
S A een
In the ease where the three rotations are chosen as parameters this follows also
from the fact that differential rotations a,, a, and a; change the vector X0 +1 by vectors
i XX, a, j X X°,4 109 and k X X°,4 103, respectively.
