to movement in three dimensions 



163 



and enable us to infer that the matrix of the homography has the 

 form 



where a, b, c, d, a', h' , c , d' are written respectively for 



^^3' ^3' ^3' ^3' ^^4' ^4? ^4j ^4' 



If then we put 



m = 



h- 



H' = 



we have 



0, 

 0, 

 0, 



h + do 



0, 



h-t 



0, 

 0. 



m" 



m, 0, 

 0, 

 0, 

 0, 



0, 

 0, 



9, 

 1, 

 0, 



h + ^0, 0, 

 0, h~t 

 bm. 



am' 



a'm~^, b'm, 



f, 

 9, 

 c, 

 c'. 



V 



u 

 d 

 d' 



If the transformation associated with the matrix /x change the 

 point {x, y, z, t) to (xj, y-^, z-^, ^i), so that (xj) = /x (x), we have from 

 {x') = ^ (x), for the change from {xj) to («'), the equations expressed 

 hy {x') = ^/x-i (xj), which, by what we have just seen, are the 

 same as 



{h + Oq) x'+ fz'+ vt'= {h + ^o) «i + A + '^h' 



{h - do) y'+ gz'+ ut'= {h - ^o) «/i + 9H + ^h> 



z'^ am~^x^ + bniy^ + czj^ + dt^, 



t'= dm-^x-^ + b'my-^ + c'z^ + d't-^. 



Without entering into general statements about the minors of a 

 matrix, we proceed now to show in detail in an elementary way 

 that these last are the equations of what we have called an axis- 

 range perspective. For this purpose, write 



F = 



f 



7 = 



, G = 



9 



U 



h + do' A + ^o ^-. 



C = c — am-'^F — bniG, D = d — ayn-W 



C'= c'- a'm-'^F - b'mG, D'= d'-a'm-W 



h-d, 



- hmU, 



- b'mU, 



11—2 



