affine conics

In affine geoemtry, we do not have the concept of length, or even of the ratio of
non-parallel segments. It follows that we cannot use the focus-directrix definition.
We therefore define conics in affine geometry as the loci with equations quadratic
in x and y, other than the degenerate cases.

First, we must check that this does give a class of affine objects, this is done in

Theorem AC1
If C has equation f(x,y) = 0, with f quadratic in x and y, and C is
not degenerate, then, for any affine transformation t,
(1) t(C) has equation F(x,y) = 0, with F quadratic in x and y, and
(2) t(C) is not degenerate.

proof of Theorem AC1

In the study of plane conics in euclidean geometry, we saw that a parabola has
a single reflection symmetry, but that an ellipse or hyperbola has two, with axes
at right angles. The product of these is a half-turn (rotation through π) about the
point of intersection. Although we cannot talk of angle in affine geometry, any
element of E(2) is a member of A(2). Thus half-turns are affine transformations.
They turn out to be very useful in the study of plane conics in affine geometry.

half-turns in affine geometry

In euclidean and similarity geometry, there are infinitely many congruence classes
of plane conics. With the larger group A(2), we might expect fewer classes. This is
true - there are only three classes!

affine symmetry groups of plane conics

In euclidean geometry, we saw that the symmetry group of a plane conic C is of
order two if C is a parabola,and of order four otherwise. The affine symmetry
group of C is always infinite. We know that congruent figures have conjugate
symmetry groups. Since there are only three affine classes of conics, there are
essentially only three symmetry groups. These turn out to be quite different,
giving a method of proving that there really are three distinct affine classes of
plane conics.

classification of affine conics

Finally, we consider the concept of tangency. We have to show first that it is an
affine concept, i.e. that if L is a tangent to a conic C and t is an affine transformation,
then t(L) is a tangent to t(C). We can now apply our affine knowledge to discover
interesting results about every conic by looking at just three cases!

tangents in affine geometry

main affine page

main conics page