newson: projective transformations. 



59 



and 



( 21) 



24. The Explicit Normal Form of S. If the above equations 

 ■of the transformation S are solved for x, and y,, we get the 

 following explicit normal form: 



and y,: 



(22) 



In homogeneous coordinates these may be written in the form 



X }' z o 



A B C A 

 ^^1 A, B, C, kA, 

 I p o k 



X }• z o 

 __\A B C B 

 Py^-IA, B, C, kB, 

 I p o kp 



pz, 



X y z o 

 A B C C 

 A, B, C, kC 

 I p o o 



(23 



§5. Type V; Hlations, Properties and Normal Form. 



25. A Single Characteristic Constant a. In the case of an 

 elation the invariant figure consists of all points on a line I and all 

 lines through a point O on 1. An elation S' transforms a point P 

 of the plane into P, some other point on the line OP. On each of 

 the invariant lines through O there is a one-dimensional parabolic 

 transformation having its single invariant point at O. Ever}- 

 pencil of lines having its vertex at A, some point on 1, is an invari- 

 ant pencil of the transformation. In each of these invariant pen- 

 cils there is a one-dimensional parabolic transformation having I 

 for its single invariant line. 



