Coxeter- Section 9.5 (2)

Pinyi Yao
As discussed in Section 9.5 (1), in projective geometry, degenerate conics would be two lines. In the construction below, we are omitting the phrase "but not x' is perspective to y'" from the statement of Steiner's construction 8.51. We build a perspectivity. Point P and Q are pencil points, and line o is the range. R is a point used to build line x', and we'll keep it there in order to move line x' around. O=x'·y'.(We turn trace on for point O.) We have the perspectivity: x' is perspective to line y'. Move point R around (same as move line x' around), and see the trace/locus of point O. What do you see? You should see the locus of point O is line o. You might wonder that's only one line, where is the other line? What if line x' and y' coincide? In other words, what if point O lies on line PQ? In that case, all the points on line PQ would be points to make x' perspective to y'. So PQ is our second line. We have two lines, o and PQ, to be the degenerate conic.