We know that the euclidean symmetry group of a parabola has order two, while those
of an ellipse and hyperbola have order four (and that the latter each contain a half-turn).
We also know from the general theory of symmetry that if two figures are congruent
in the geometry, then their groups are conjugate. Thus, here, it is enough to determine
the affine symmetry groups of the three standard conics.
Also from the general theory, we know that the affine symmetry group of a plane figure
contains the euclidean symmetry group as a subgroup. As we shall see it is not always a
proper subgroup.
The details for each type of conic are rather different, so we have separate pages.
The proofs are not particularly geometric, and could well be ignored.
In each case, we find that the symmetry group is infinite. The actual nature of the groups
is not particularly important (except as a means of distinguishing the conics - see below).
There is one feature which is, however, significant. It leads to the following general result.
The Affine Transitivity Theorem
If P and Q are points on a conic C, then there is an affine transformation which
maps C to C and P to Q.
Proof
From the results for individual types of conic, there is an affine transformation u
which maps C to the appropriate standard conic C0,
and P to a particular point R
on the standard conic. Similarly, there is a transformation v mapping C to C
0,
and Q to R.
Then vou-1 maps C to C, and P to Q, as required.
This has important consequences. In euclidean geometry, we can identify as special the
point or points where the axis of symmetry through a focus meets the conic. For example,
the vertex of a parabola. The theorem shows that this can be transformed to any point
on the same conic. Thus, there is no affine characterization of the vertex. A similar
argument shows that the major axis of an ellipse, and the transverse axis of a hyperbola
are not affine concepts. It also confirms that the foci do not have affine descriptions,
since they lie on these axes.
We can also use the groups to show that no two of the three standard conics are affine
congruent. First of all, we observe that the symmetry groups of affine congruent conics
are conjugate. The symetry groups of ellipses and hyperbolas contain a half-turn, and
a conjugate of a half-turn is a half- turn. The symmetry group of P0 does not contain
any half-turn (on geometrical grounds if nothing else, since a half-turn would make the
parabola "point" the other way). Thus a parabola is not affine congruent to an ellipse or
hyperbola. The symmetry group of an ellipse contains elements of all finite orders, but
the group of a hyperbola has only elements of order two. Thus an ellipse cannot be affine
congruent to a hyperbola.
There are also applications to the problem of tangents in affine geoemtry.
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