What is the Canonical Form of a Conic?

Conics. A conic is any planar curve described by a second order polynomial in x, y, equated to 0. Say:

     + xy + + x + y + = 0,      provided: (, , )  ≠ (0,0,0).
     

Canonical form.  Non degenerated conics are divided into ellipses, parabolas, and hyperbolas.
An ellipse or an hyperbola is in canonical form if:
    (1) its main symmetry axis (the one including the two foci) is  the x-axis.         
    (2) its center is in the origin of the axis.
A parabola is in canonical form if:
    (1) its main symmetry axis (the one including the focus) is  the y-axis.         
    (2) its vertex is in the origin of the axes.

Any Conic has a canonical form. Any conic may be transformed into canonical form by:
    (1) first rotating it, in order to make its main symmetry axe horizontal (vertical),
    (2) then translating it, in order to put its center (its vertex) into the origin.
We will shortly sketch a method for computing the rotation and the translation, then write a program turning (almost) any conic into its canonical form.

Created by Mathematica  (August 4, 2004)