The Imaginary hi Geometry 



13 



the origin and has the same length. If the red vector be taken 

 for a velocity then the equation represents a rotation of the entire 



/ 

 v 



Fig. 14. 



plane about the origin as center at the angular rate positive unity. 

 When y = ix then y = — ix, so that the conjugate vector belongs 

 to the conjugate line y = — ix. 

 The lines 



(y — b) = i(x — a) and (y — b) = — i '(■* — a), 



also conjugate, differ from the foregoing only in having their 

 center at (a, b). The pair together form the point circle 



(x— a y+(y— by 



o. 



All such pairs of lines cut the line at infinity 1 in the same two 

 " circular vectors at infinity." We call each line a " circular ray." 

 Moreover we denote the vector at infinity when the slope is -\- i 

 by /, when — i by J. 



Every other central line is elliptic, is a parallel projection of 

 a circular ray. 



It suffices to prove this for 



y = re i *x. 



If we multiply both sides of the equation by e l9 it remains satis- 

 fied, i. c, c ie (x, y) satisfies the equation if (x, y) does. But 

 we have already seen that the red vector e' e (x, y) is tangent to 

 the same ellipse that (.r, y) is and that its blue point is on (the 

 'To fix the ideas we think of x = 00 as the line at infinity. 



13 



