THEORY OF POLYCOXIC PROJECTIONS. 



53 



live circle. The polar distance of a circle is the an^ar 

 distance of any point of its circumference from either of 

 its own poles. The inclination of a circle is the angle 

 between its plane and the primitive plane. It is meas- 

 ured by the arc distance between the pole of the given 

 circle and the pole of the primitive circle, smce this 

 measures the angle between the perpendiculars to the 

 planes of the two circles. 



In figure 14 let NESW be the primitive circle and let 

 QR be the trace of the plane of a small circle, with P as 

 its pole; then FR = PQ is its polar distance and PN is its 

 inclination. The diameter WE is called the line of measures 

 of the circle QB; NS is perpendicular to WE at the center 



Tig. 15.— Determination of the arc distance from tlxe center on stereographic projection 



of the primitive circle. S is the point of projection and 

 Q' and R' are the projections of the extreme or principal 

 elements of the oblique circular cone SQR which is formed 

 by the projecting lines of the pouits of the circle QR. 

 Denoting the polar distance of the circle by k and the 

 inclination by ^, we. have 



OR' = a tan ^('^ 



k) 



0Q'=atan-2(« + ^) 



Problem 1. — ^To determine the shortest distance between 

 the center of the map and another point the projection of 

 w^hich is given; that is, to determine the arc of a great 

 circle between them : 



