central conics revisited

In the theory of plane conics, we have the following

Theorem
If C is a plane conic and F is a family of parallel chords of C,
then there is a line L which bisects every member of F.

The statement involves the concepts of conic, parallelism and bisection,
so it is natural to regard it as an affine theorem. Indeed, it can be proved
as such, but we have to treat the ellipse, parabola and hyperbola as
separate cases.

Here, we shall prove it from a projective standpoint. This also gives a
new insight into the idea of the centre and diameters of a plane conic.

 Suppose that C is a projective conic, i.e. a p-conic. In the model RP2, C is represented by a cone with vertex O. If Π is a plane not throught O, then we can embed RP2 in Π. The plane Π* through O, parallel to Π embeds as the ideal line for Π. Each direction in Π* embeds as an ideal point for Π. The p-conic C embeds as the plane conic C(Π) = CnΠ, and the ideal points corresponding to CnΠ*. In RP2, Π* represents a p-line L(Π), so meets C at most twice, and we know that C(Π) will be an ellipse if there are no intersections, a parabola if there is just one, and a hyperbola if there are two. Also, any plane conic C' arises as the non-ideal part of such an embedding, i.e. as C(Π) for some p-conic C and plane Π. Indeed, we can find a suitable p-conic for any Π not containing O. This is described in more detail in a separate page. Definitions Suppose that C is a p-conic, and that π is a plane not through O. (1) The Π-centre of C(Π) is the embedding in Π of the pole of the p-line L(Π) with respect to the p-conic C. (2) A line in the embedding on Π is a Π-diameter of C(Π) if it contains the Π-centre. We will have to check later that these definitions depend only on C(Π), and not on the choice of C and Π. Note that the definitions allow for cases where the Π-centre is an ideal point for Π. Although we try to present our results for all plane conics, this possibilty does lead to some problems. We make the following Observations With the above notation, the Π-centre is ideal for Π if and only if the plane conic C(Π) is a parabola. In such a case, (1) the p-line L(Π) is the tangent to C at the Π-centre, (2) each Π-diameter cuts C(Π) exactly once, and (3) the Π-centre corresponds to the direction of the axis of C(Π). These are quite easy to see. Notice first that the ideal points for Π correspond to the p-points of L(Π). Hence the Π-centre will be ideal for Π if and only if P, the pole of L(Π) lies on L(Π). In such a case, L(Π) is the tangent to C at P, and hence P lies on C. It follows that any other p-line through P cuts C exactly once, so the embedding cuts C(Π) once. As P', the embedding of P, is the Π-centre of C(Π), and is ideal for Π, it represents a direction on Π. Each line in this direction cuts C(Π) once. This situation arises only for a parabola, and the direction must be the axial direction. The Π-diameter property If A',B' (A'≠B') on C(Π) are such that A'B' is a Π-diameter of C(Π), then the Π-centre of C(Π) is the mid-point of A'B'. Since we know that such chords exist only when C(Π) is an ellipse or hyperbola (i.e. is a central conic), it is reasonable to guess that the Π-centre will be the centre as previously defined. This will follow from our next result. But the result is mainly of interest as the unified version of the classical theorem stated at the top of the page. The parallel chords theorem If F is a family of (at least two) parallel chords of C(Π), then there is a unique Π-diameter which bisects every member of F. As remarked earlier, a given plane conic C' may be described as C(Π) for (infinitely) many choices of Π and C. We now show that, for any choice, the Π-centre will be the same (possibly ideal) point. We do this by showing that the point is determined by the affine geometry of C'. The independence theorem (1) If C(Π) is a parabola, then the Π-centre is the ideal point for Π corresponding to the axial direction for the plane conic. (2) If C(Π) is an ellipse or hyperbola, then the Π-centre is the affine centre of the plane conic. Part (1) has already been established in the above observations. With this result, we can now refer to the centre and diameters of any plane conic, even a parabola.