newson: projective transformations. 



51 



P h Ph 



- ' — ! ^--=^cos<^-f cot^i sin^ — (cos<^-f cot^ sin<^) 



(cot^, — cot^) sin<^=a sin<^. 



(10) 



.-. in^-a sine/) and is constant for all pairs of corresponding points 

 in the transformation T'. 



II. Implicit Normal Form of the Transformation T'. The 



two formulas just proved enable us to write down the analytic 

 expression for the transformation T' in its implicit normal form. 

 Let the coordinates of A be (A,B), of B be (AjB,) and let p be 

 the tangent of the angle which the line Al makes with the axis of 

 X. In the formulas 





we can substitute for these perpendiculars their analytic expression 

 in terms of coordinates of the invariant points and the inclination 

 p of the lines 1 and 1'. Making these substitutions we have, 



X y I 

 A B i: 



I P o 



A,B, 1 

 I p o 



X y I 

 AjB, I 

 I p o 



^1 Yi I 

 A B I 



A, B. I 



=k 



1 i+p- 



X \ 

 A B I 

 A B, I 



and 

 I 



a sin<^. 



Since | i -|-p- = seci/', d is distance AB. and a 

 reduce to 



sin(^ 

 AB' 



these 



^1 Yi I 

 jA B I 

 A. B, I 



X y I 

 A B I, 

 A, B, li 



md 



X y I 

 A>B, I 

 I p o 



X, y, I 

 A B I 

 A, B. I 



X y I 

 A B I 

 A, B, I 



COSv/' 



(II) 



12. Explicit Norma: Form of Type II. The above equations 

 are both linear in x^ and y, as may readily be seen by expanding 

 the determinants. Expanding and solving for Xj and y, we get 



