56 



KANSAS UNIVERSITY gUARTERI.Y. 



X V I O 

 A B I B 



I p o Baa-f-p 



Pi O B/~a2— h'tt Wp«+P, 



with the condition that 



h'a 



(17) 



In homogeneous coordinates these may be written 



px, 



py. 



pz, 



X y z o 

 A B C A 

 I p o Aaa-|-i 



p, O A| a- — h'uj-f-a+- 



X y z o 



A B C B 



I p o Baa-j-p 



I 



p, O b/^ — a2 — h'aW 



P« + Pi 



X y z o 

 A B C C 



I p o Caa 



p^ O C[ tt^ — h'al 



(18) 



>J4. 



Type 4: Perspective Transformation, Properties and 

 Normal Form. 



19. Single Cross-Ratio k. The fundamental invariant figure 

 of a perspective transformation consists of a line 1, a point O not 

 on 1, and the pencil of rays through O. In the special case that 

 the point O lies on 1, the transformation becomes an elation. 

 Fixing our attention on the fundamental invariant figure we see 

 that every invariant line in the plane except 1 has on it two invari- 

 ant points, O and its point of intersection with 1; also every pencil 

 having its vertex A on 1 is an invariant pencil in the plane, and 

 has in it two invariant rays, 1 and the line AO. The effect of a 

 perspective transformation in the plane is to move a point P along 

 the invariant line PP^ to P^. Thus we have on each of the invariant 

 lines through O a one-dimensional hyperbolic transformation with 

 two invariant points O and A. Likewise in each of the invariant 



