### COURSE OF PROJECTIVE GEOMETRY

§ 23: Conics of points and conics of lines

Definition: A non-degenerated conic of points is the set of intersection points x . x' , where xx' is a projectivity (but not a perspectivity) from a pencil of lines L onto a pencil of lines M (with LM).

Note: Each projectivity φ : LM is the restriction to L of a projective transformation of P2. So this definition is equivalent to the one in §21.

Definition: A non-degenerated conic of lines is the set of connecting lines XX ' , where XX ' is a projectivity (but not a perspectivity) from a pencil of points l onto a pencil of points m (with lm).

Note: It will turn out (in §25) that a conic of lines is the set of tangents to a conic of points.

Example:
Given a, b, c through L and a ' , b ' , c ' through M. Construct the points of the conic that corresponds to the projectivity from L to M which maps a to a ' , b to b ' , and c to c ' .
Solution:
First construct the Pappus point P of the projectivity.
Let x be a line through L. Construct x' using the construction of Steiner. See O45.
The intersection point x . x' is a new point of the conic.
Note: (See O43.)
If x = LM then x ' = MP so x . x' = M.
If x = LP then x ' = LM so x . x' = L.
So L and M belong to the conic.

Example:
Given five freely situated points (that is to say: no three collinear). Construct a sixth point of the conic through these five points.
Solution:
Call two of these points L and M, and the other three A, B and C.
Let a := LA, b := LB, c := LC, and let a ' := MA, b ' := MB, c ' := MC.
Let φ be the projectivity from L to M with φ(a) = a ' , φ(b) = b ', φ(c) = c '.
Then φ generates the required conic.
Construct a sixth point as in the last example above.

O84 Investigate which degenerations occur if in the definition of a conic of points L = M, or if the projectivity is a perspectivity. Likewise for a conic of lines.

O85 Let A, B and C be points on a line l and let A ' , B ' and C ' be points on a line m.
Let φ be the projectivity from l onto m with φ(A) = A ' , φ(B) = B ', φ(C) = C '.
Construct the lines of the conic of lines generated by φ.
Furthermore, prove that l and m belong to this conic of lines.

O86 Given five freely situated lines. Construct a sixth line of the conic of lines determined by these five. (For a good picture: start from five tangents to a circle.)

O87 Prove a line can't have three distinct points in common with a non-degenerated conic (of points).