The general 2-variable quadratic equation can be written

(1) |

(2) | |||

(3) | |||

(4) | |||

(5) |

Then the quadratics are classified into the types summarized in the following table (Beyer 1987). The real (nondegenerate) quadratics (the Ellipse, Hyperbola, and Parabola) correspond to the curves which can be created by the intersection of a Plane with a (two-Nappes) Cone, and are therefore known as Conic Sections.

Curve | ||||

Coincident Lines | 0 | 0 | 0 | |

Ellipse (Imaginary) | ||||

Ellipse (Real) | ||||

Hyperbola | ||||

Intersecting Lines (Imaginary) | 0 | |||

Intersecting Lines (Real) | 0 | |||

Parabola | 0 | |||

Parallel Lines (Imaginary) | 0 | 0 | ||

Parallel Lines (Real) | 0 | 0 |

It is always possible to eliminate the cross term by a suitable Rotation of the axes. To see this, consider
rotation by an arbitrary angle . The Rotation Matrix is

(6) |

so

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) |

Plugging these into (1) gives

(12) |

(13) |

(14) |

(15) |

(16) | |||

(17) | |||

(18) | |||

(19) | |||

(20) | |||

(21) |

The cross term can therefore be made to vanish by setting

(22) |

For to be zero, it must be true that

(23) |

(24) |

(25) |

(26) |

(27) |

Rotating by an angle

(28) |

(29) |

(30) |

(31) |

(32) |

(33) |

(34) |

Consider an equation of the form
where .
Re-express this using and in the form

(35) |

(36) |

(37) |

(38) | |||

(39) | |||

(40) |

Therefore,

(41) | |||

(42) |

From (41) and (42),

(43) |

(44) |

so

(45) |

(46) | |||

(47) |

Adding (46) and (47) gives

(48) | |||

(49) |

Note that these Roots can also be found from

(50) |

(51) |

(52) |

(53) |

(54) |

Let be any point with old coordinates and be its new
coordinates. Then

(55) |

(56) | |||

(57) |

If and are both , the curve is an Ellipse. If and are both , the curve is empty. If and have opposite Signs, the curve is a Hyperbola. If either is 0, the curve is a Parabola.

To find the general form of a quadratic curve in Polar Coordinates (as given, for example, in Moulton 1970), plug
and into (1) to obtain

(58) |

(59) |

(60) |

(61) |

(62) |

Using the trigonometric identities

(63) | |||

(64) |

it follows that

(65) |

Defining

(66) | |||

(67) | |||

(68) | |||

(69) | |||

(70) |

then gives the equation

(71) |

(72) |

(73) |

**References**

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 200-201, 1987.

Casey, J. ``The General Equation of the Second Degree.'' Ch. 4 in
*A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing
an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl.* Dublin: Hodges, Figgis, & Co., pp. 151-172, 1893.

Moulton, F. R. ``Law of Force in Binary Stars'' and ``Geometrical Interpretation of the Second Law.'' §58 and 59 in
*An Introduction to Celestial Mechanics, 2nd rev. ed.*
New York: Dover, pp. 86-89, 1970.

© 1996-9

1999-05-25