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2
2. The Invariants of the Orientation.
The relative orientation of a pair of photographs is defined in an
earlier paper (1) in terms of the base components (C, Cy, C,) and
the Eulerian angles ? v and $.
The condition of correct relative orientation gives the relationship
y (C, 5— C, 5) +7 (x Cz+ C, f) -— Cy (x$ Ef) 0. .. (1)
where (x, y) are the coordinates of the image in the left hand picture ;
(8,7,5) are given in terms of the photo-coordinates (x', y') in the
right hand photograph by the transformation matrix
— L sin + cosa M sin y — sinA . — sin? ibt
L cosy + sinA —M cosy + cosa sing cosy A2)
sin '&- sin $ — sind cos ¢ cos 2-
where
L = (1 —cos®) sing , M = (1-—cos#)cos@ … .. (3)
and
A = y $ - the difference between the azimuthal angles.
The value of ^ will usually be small when the principal-point bases
are employed as x axes. The actual value can be derived by applying
condition (1) to the principal points P: and P ET follows :
2
Hence,
»7/ — (fsin?sinp, —fsin®cosy, —f coss).
(£5756) P, (f sin® sin sin Ÿ cosy cos 4 )
Substituting in Equation (1) we obtain.
P [zz 5T 922 5 E Yezeej).
C C. bCz.. ,b
y/ x = re sin 9 cosy Av cos? — sin? siny ) … .. (4)
P2(X5y202--f; X= - bb; y = 2 = 0)
Hence,
(5,7 3:5) P,
( fsin? siny — b ( — L sinyy + cosa ),
— f sin 9 cosy —b ( L cosy + sina),
— f cos $- — b sin$ sing)