THEORY OF POLYCONIC PROJECTIONS. 



59 



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Fig. 20.— Projection of a great circle through the projections of two given points on stereo- 

 graphic projection. 



In figure 20 let ORG'S be the primitive circle and let 

 P and Q be the projections of the two given points, and 

 let Jl be the center of the projection. The lines that pro- 

 ject any two antipodal points are perpendicular to each 

 other; we can then easily determine the projections of 

 the points antipodal to P and Q through which the pro- 

 jected circle must necessarily pass. Draw PA and prolong 

 it beyond A; at A erect the perpendicular AOj intersecting 

 the primitive circle at 0; draw OP and erect upon it the 

 perpendicular OP' intersecting PA produced in P'-, P' is 

 then the projection of the point antipodal to P. The tri- 

 angle OPP' is the projecting triangle turned on the pro- 

 i'ected line PP' as an axis into the plane of the paper, 

 n a similar way Q' can be determined, but a circle passed 

 through P, Q, and P' is the required projection. It may 

 be seen that the construction is correct from the considera- 

 tion that J-P' must be a third proportional to AP and AO, 

 If the point of which P is the projection has the polar dis- 



