One of the most beautiful parts of plane geometry is the study of conics.
These were studied by the Greeks. It was Apollonius
ellipse, parabola and hyperbola for the three different types.
Plane conics may be defined in three ways:
We will see that each approach leads to essentially the same family of curves.
- as loci satisfying the focus-directrix property,
- as loci with equations quadratic in x and y,
- as plane sections of a right-circular cone.
The circle, a special kind of ellipse, does not have a focus-directrix definition,
but appears naturally in the other approaches.
We shall look at the congruence classes of conics in various geometries. In
particular, we will find that the infinite number of classes in euclidean and in
similarity geometry reduce to just three classes in affine geometry, and to a
single class in projective geometry.