Special case 



In the special case (fig. 3) the following 

 definitions should be noted: 



1 . Point A is the optical center of the 

 lens of a first camera . 



2. Point B is the optical center of the 

 lens of a second camera . 



3. Line AB is parallel to the line HJ. 



4. Point O is the reference point from 

 which all measurements of distance 

 on the line HJ will be taken. 



5. Point P is a point on the fish. 



6. Point P lies in the plane HABJ, sub- 

 ject only to the simplifying restriction 

 that OF is positive (that F lies to the 

 right of O) . 



In a picture taken by camera B, point P 

 will appear to be at point F on line HJ. Similar- 

 ly, in the picture taken by camera A, point P 

 will appear to be at point G on line HJ. Also, 

 we have made D the perpendicular projection of 

 A on the line HJ. Tlie problem now is to derive 

 an e quation in which we can use the distances 

 OF and OG to obtain the unknown distance OQ. 

 From the fact that figure 3 contains a number 

 of similar triangles, our required equation is 

 easily derived geometrically as follows: 



Since triangles ABP and PFG (fig. 3) are similar, 



then PC = (FG)(PT). (1) 



but, 

 so, 



PC = (FG)(PT) . 



AB 



PT = AD - PO, 



(2) 



PQ = (FG)(AD-PQ) (3) 



AB 



simplifying PQ = (FG)(AD) (4) 



FG+AB 



Consider now the similar triangles AEXj 



Then 



or. 



QG = OG - OP 



PO AD 



(5) 



QG = PQ ( OG - OD ) (6) 

 ( AD ) 



now, QG = OG - OQ, (7) 



and substituting (4) and (7) into (6) we have 



OG - OQ = ( (FG)(AD) ) ( OG - 0D) (8) 

 ( FG + AB) ( AD ) 



Then OQ = OG - ( ( FG)(AD ) ) ( OG-OD ) (9) 

 ( FG-I-AB ) ( AD ; 



This reduces to OQ = OG (AB + OD) - (OD) (OF) 



AB-l-(OG - OF) (10) 



which locates the perpendicular projection of P 

 on the orientation line HJ. 



F Q C G 



andPQG. 



The distances AB and OD are constants 

 determined by the positions of the cameras in 

 relation to point O on line HJ and to each other. 

 The distances OF and OG are variables deter- 

 mined by the position of P in the plane HABJ. 

 The value OF can be determined from a photo- 

 graph taken by camera B, and that of OG from 

 one taken by camera A. 



General case 



In the gene ral c ase (fig. 4), where we 

 wish to determine OQ when P lies anywhere in 



