Desargues' Brouillon Project and the Conics of ... - J.P. Hogendijk

18 Jan I! Hogendijk
r2. Therefore L is the pole of q, which had to be proved. Desargues
derived (A,) and (A2) from a more exciting and more general
theorem, as we will see below.
We will now show that in modern terms pole and polar are
projectively invariant concepts. In order to make this clear, we
suppose that planes Vand V’ intersect a cone with apex Tin a circle
rand a conic section r’ (Figure 10). For any point P in V; call the
projection of P the intersection P’ of TP with V’, as in Section 2.
Then the projection of a line p in V is a line p‘, which is the
intersection of V’ and the plane through Tand p. The projection of
r is of course r‘. Now suppose that P is the pole of line p with
respect to r. Let PXQY be a straight line which intersects r at X
and Yand p at Q. The projection of PXQ Yis a straight line P’Q’X’ Y’
which intersects f ‘ at X’ and Y’ and p’ at Q’. Because P is the pole
of p, the pairs P, Q, and X, Yare four points in involution. By a
Figure 10.
T