Suppose that C is a projective conic, i.e. a pconic.
In the model RP^{2}, C is represented by a cone with vertex O.
If Π is a plane not throught O, then we can embed RP^{2} 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 pconic C embeds as
 the plane conic C(Π) = C_{n}Π, and
 the ideal points corresponding to C_{n}Π*.
In RP^{2}, Π* represents a pline 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 nonideal part of such an embedding,
i.e. as C(Π) for some pconic C and plane Π. Indeed, we can find a suitable
pconic for any Π not containing O.
This is described in more detail in a separate page.
Definitions
Suppose that C is a pconic, and that π is a plane not through O.
(1) The Πcentre of C(Π) is the embedding in Π of the pole of the pline L(Π)
with respect to the pconic C.
(2) A line in the embedding on Π is a Πdiameter of C(Π) if it contains the
Πcentre.
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 Πcentre 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 Πcentre is ideal for Π if and only if the
plane conic C(Π) is a parabola. In such a case,
(1) the pline L(Π) is the tangent to C at the Πcentre,
(2) each Πdiameter cuts C(Π) exactly once, and
(3) the Πcentre 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 ppoints of L(Π).
Hence the Πcentre 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 pline through P cuts C exactly once, so the embedding
cuts C(Π) once. As P', the embedding
of P, is the Πcentre 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 Πdiameter property
If A',B' (A'≠B') on C(Π) are such that A'B' is a Πdiameter of C(Π),
then the Πcentre of C(Π) is the midpoint of A'B'.
proof
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
Πcentre 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 Πdiameter which bisects every member of F.
proof
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 Πcentre 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 Πcentre 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 Πcentre is the affine
centre of the plane conic.
Part (1) has already been established in the above observations.
proof of part(2)
With this result, we can now refer to the centre and diameters of any
plane conic, even a parabola.
