three-dimensional space in view of the two 

 cameras, the following definitions should be 

 noted: 



1. In this three-dimensional drawing, 

 line HJ lies on the line formed by the intersec- 

 tion of the perpendicular planes 1 and 2. 



2. Line OO' lies in plane 2 and is per- 

 pendicular to line HJ. 



3 . Line AB lies on the line formed by 

 the intersection of planes 1 and 1' . 



4. Plane 1 in figure 4 is the same as 

 the plane HABJ in figure 3, except that lines 

 AD and BC have been omitted to simplify the 

 drawing. 



5. Plane 1' is in the position plane 1 

 would occupy if, using the line AB as an axis, 

 plane 1 were rotated through some angle, say 

 alpha d(_ (subject to the obvious restriction 

 that point P' will lie in the field of view of the 

 two cameras) . 



6. Point P' is the position that point P 

 would occupy if plane 1 were rotated through 

 the angle alpha, and point Q, therefore, is the 

 location of the perpendicular projection of the 

 points P and P' upon line HJ. 



7. Point F' is the position that point F 

 would occupy if the line FP were extended as 

 plane 1 rotates through the angle alpha; points 

 G' and H' bear similar relationships to points 

 G and H. From a consideration of figures 3 and 

 4, we see that the derivation of equation 10, al- 

 ready given, applies to the special case where 

 the angle alpha is equal to zero. 



Since P is any point in plane 1, the prob- 

 lem of locating P anywhere in space (anywhere 

 on the fish) now becomes the simple one of show- 

 ing that equation 10 applies for all values of 

 alpha . 



This proof depends upon the fact that 

 changing the value of alpha does not change any 

 of the angular relationships in plane 1; for ex- 

 ample, although triangle P'F'G' is larger than 

 triangle PFG for all values of alpha different 

 from 0°, the corresponding angles in these two 

 triangles are identical in size . Similarly, all 

 the other angles in the triangles employed in de- 

 termining equation 10 also remain constant in 

 size as the angle alpha varies . Therefore, since 

 the derivation of equation 10 depends upon the 

 similarity of the triangles and not upon the size 

 of them, this equation holds for any value of alpha; 

 in short, for any location of P in three-dimension- 

 al space. From equation 10, we therefore easily 

 obtain O'Q' . From figure 4, it is apparent that 

 O'Q', is equal to OQ, the location of the perpen- 

 dicular projection of P' on line HJ. 



rolAt Usfc la ■ayvbin ibovt plan* 2 and U*I to iB 



Figure J.— An exampl* ot ttte error caused tj p«rapeciWe; In this picture, 

 the site of the ulaon la sr««ter tban that of the flabln^ 

 veasel because tbe aalann waa cloaer to tbe caBera vban tbe 

 picture waa taken. 



