NKWSON: PROIECTIVE TRANSFORMATIONS. 



47 



coordinates of P be (x, y) and of P^ be ^^x,,y,); let the coordinates 

 of the three invariant points be (A,B), (A,,B^), and (A^jB.,). 

 We proved in Art. 4 the following equalities: 



_APAC AP^AC APBC APjBC 



'^~ APBC AP , BC ' '-y "" APAB ■ AP,AB " 



APAB AP,AB 



"^ ABAC APjAC 



These may be written in another form, as follows: 



AP.BC 



= k. 



APBC AP,AB 



APAB AP,AC 



-k. 



(3) 



APAC 

 APAB" 



AP^AC "'■ APAC APjBC ^ APBC APjAB 



Expressing the areas of these triangles in determinant form in 

 terms of the coordinates of their vertices we have 



x^ y, I 

 A, B, I 

 A., B„ I 



X y I 

 'A, B, i! 

 !A., B., I 



(4) 



These three forms are not independent and the last may be 

 regarded as superfluous. 



Putting 



follows: 



:k, and k..-r= k, then the most convenient form is as 



X y 

 A B 

 A., B. 



'^1 Yi I 

 A, B, I 

 A., B. I 



X v I] 



A, B, li 

 A.. B., I 



^1 Yi I 

 A, B, I 

 A., B„ 1 



X \- 1 



A B I 



At B, I 



X y I 



A, B, I 



A., B., I 



(5) 



These normal forms are capable of another interpretation; the 

 values of the determinants are proportional to the perpendicular 

 distances from Pj and P to the sides of the invariant triangle. 

 They express the fact that the cross-ratios of these perpendiculars 

 are constant for all pairs of corresponding points. 



