newson: projective transformations. 57 



pencils with vertices on 1 we have a one-dimensional hyperbolic 

 transformation. A one-dimensional hyperbolic transformation is 

 characterized by the constant cross-ratio of the invariant ele- 

 ments and every pair of corresponding elements. Thus along 

 the line AO we have (AOPP,)=k. Let Aj be a point of 

 1; in the pencil with vertex at A^ we have the cross-ratio 

 A, (AOPP/VT=k. Since every line through O cuts this pencil in a 

 range having the same cross-ratio k, it follows that the one- 

 dimensional transformations on all lines through O are characterized 

 bv the same constant k; also the one-dimensional transformations 

 in all pencils with vertices on 1 are characterized by the same value 

 of k. 



Theorem 4. A pcrspeciivr tratisjcuiitatioii S of tlie plane is eom- 

 pletelx eharacterized by its fundamental figure and a eliaraeteristie 

 eross-ratio !<. 



20. Type IV a Special Case of Type I. A perspective trans- 

 formation S, /. e. a transformation of type IV, may be regarded as 

 a special case of type I. We proved for type I that k^kyk^^^i, 

 where these quantities are the characteristic cross-ratios taken in 

 the same order around the triangle. Along one side, e. g. BC, of 

 the invariant triangle we have k^— (BCXX,). If kx=i and B and 

 C do not coincide, then X and X^ must coincide and and every point 

 on the line BC is an invariant point, and every line through A is 

 an invariant line of the transformation. Thus when one of the 

 cross-ratios as k^ of type I becomes unity without B and C coin- 

 ciding, T degenerates into S, a transformation of type IV. 



21. Cross-Ratio the Same on all Lines Through A. Since 



kx = i, we have kj,=r — ; thus the characteristic cross-ratio along 



CA is the reciprocal of that along AB. Interchanging C and A in 

 the formula for k^ we get the reciprocal of ky; hence the cross- 

 ratio along AC reckoned from A to C is equal to that along AB 

 reckoned from A to B. /. e. (ACYY J = (ABZZ, )r=k,. The cross- 

 ratio of the pencil through C in k2^C(ABPP,). But every line 

 through A is now an invariant line and all the lines through A cut 

 the pencil through C in the same cross-ratio k^. Thus the trans- 

 formation S produces one-dimensional transformations along each 

 of the invariant lines through A and these one-dimensional trans- 

 formations are all characterized by the same cross-ratio k. 



