THEORY 



Since a fish is three-dimensional, all of 

 the points on the fish do not lie in the same 

 plane. Consequently, distances between points 

 on the fish are difficult to measure acurately. 

 Although many types of measurements of dis- 

 tance can be made, the present study is limited 

 to longitudinal measurements (fig. 1), as they 

 are the main type now being made by the U.S. 

 Fish and Wildlife Service in the research work 

 for the Commission. 



Referring to figure 1, a longitudinal 

 measurement between any two points (Pj and P2) 

 on a fish (or any other tliree-dimensional object) 

 is the distance between the points when they are 

 projected perpendicularly upon the axis of the 

 fish (or object). Since this axis is not access- 

 ible, the measurements are made instead 

 between their perpendicular projections {Q[ and 

 Qo) upon an orientation line (HJ), which is 

 located parallel to the axis. 



I— LD ■i 



.1 Ic 



"^=:7 



LD = Longitudinal distonce between ttie points P, and P2 



on the fish 

 HJ = Onentotion line poroliel to oxis of fish 

 = Reference point on line HJ 



Plgur« l.^-HIuatntloa of ■Bsnlog of "lAn^tudliial dlitance". 



The loo^tudloal dlit«oce b«tv««o the point! F^ and P^ oo 

 the flBh Is eiiual to ^2 ~ ^ OD the orleotetloD line HJ. 



Photography offers a ready means of 

 projecting P^ and P2 upon the orientation line 

 to obta in the points Q^ and Q9. The distance 

 Q.Q2, however, cannot be measured directly. 

 Instead, each point on the line is located in 

 terms of its distance from some reference 

 point, say 0, on the line HJ; that is, the location 

 of the perpendicular project ion of P^ on the 

 orientation line HJ is OQ, . and the location of 

 the perpendicular projection of P„ is OQTT 

 Then OQ^ = OQ^ - Q^Q2 - LD. 



In locating the perpendicular projection 

 of some point in space, say P^, upon the orienta- 

 tion line by photography, we find there are two 

 sources of error: (1) parallax and (2) perspective. 



Parallax Error 



An example of parallax is shown in figure 

 2. In this figure, point P, which can be any 

 point on die fish, is shown in three different 

 positions above the line HJ. In each case, Oi. 

 Q„, or Qo represents the location of the per- 

 pendicular projection of P upon this line. In only 

 the first case, where P is directly beneath the 

 camera, will Q appear to coincide with P as 

 viewed by the camera. In the other cases, 

 points P„ and P3 will appear to be located at 

 P^ a nd Po , as illustrated. The distancesP 9Q2 

 and P oQo are the piarallax errors. 



The problem of parallax can be solved by 

 the use of two cameras. To show how this can 

 be done, we have two considerations: (1) the 

 special case of determining OQ (fig. 3) where 

 P is located in the plane HABJ formed by two 

 cameras and an orientation l ine, and (2) the 

 general case of determining OQ (fig. 4) where 

 P does not lie in the same plane with the orienta- 

 tion line but is still in the field of view of the 

 two cameras . 



Posit ion 



Position 



PorollOK Error 



Porollox Error 



No Parollan Error 



Figure 2."EUBpleB of what Is aeant by parallsx error. 



