GEOMETRIC CONSTRUCTION. 79 
two planes; and the cross-ratio of any four collinear points, or 
any four concurrent lines, in the one plane is equal to the 
cross-ratio of the four corresponding points or lines in the 
other plane. Our construction is therefore fully entitled to 
be called a non-perspective projection of a into z’. 
THEOREM 8. Two conics K and K’ in the planes z and 7’ re- 
spectively, both touching their line of intersection 1, determine a 
non-perspective projection of 2 into 2’. 
94. Real Non-perspective Projection. In the special case 
that the conics K and K’ are real conics, the non-perspective 
projection transforms real points and lines into real points 
and lines. In this case the points inside of K cannot be con- 
structed directly. It is evident that the polar of P with re- 
spect to K projects into the polar of P’ with respect to K’. 
Let P be a point inside of K and let p be its polar with re- 
spect to K. Choose two points on p outside of K’ and con- 
struct the corresponding points in 2’. These points determine 
the line p’, the projection of p. Construct the pole of p’ with 
respect to K’; this point is P’, the projection of P. 
95. Four Pairs of Corresponding Lines. Let four lines 
a, b, c, d, be chosen in z and their four corresponding lines 
a’, b’, c’, d’inz’. Let each set be so chosen that no three of 
them are concurrent and no two meet on l. The five lines 
a, b, c, d, | determine a conic K in z, and the five a’, b’, c’, 
d’, l’ determine K’ inz’. The two conics K and K’ determine 
uniquely and completely a non-perspective projection of xa 
on 7’. 
THEOREM 9. Four pairs of corresponding lines in the most 
general position are necessary and sufficient to determine a non- 
perspective projection of one field of lines on another. 
96. Four Pairs of Corresponding Points. Let us choose 
four points A, B, C, D in x and four points A’, B’, C’, D’ in x’. 
Let these points in each plane be so chosen that no three of 
them are collinear. Let us assume that A and A’, B and B’, 
etc., are pairs of corresponding points in a non-perspective 
