Real Nondegenerate | Imaginary Nondegenerate | ||

Ellipse | x^{2}+y^{2}-1=0 | Imaginary Ellipse | x^{2}+y^{2}+1=0 |

Hyperbola | x^{2}-y^{2}-1=0 |
||

Parabola | x^{2}-y=0 |

Real Degenerate | Imaginary Degenerate | ||

Intersecting Pair of Lines | x^{2}-y^{2}=0 | Imaginary Pair of Lines, Intersecting at a Real Point | x^{2}+y^{2}=0 |

Parallel Lines | x^{2}-1=0 | Imaginary Parallel Lines | x^{2}+1=0 |

Pair of Lines, one at Infinity | x=0 |

Square Degenerate | |

Coincident Lines | x^{2}=0 |

Coincident Lines at Infinity | 1=0 |

**Theorem**: In a real linear system of conics, there is at least one conic which contains a real line (so that conic is either "real degenerate" or "square degenerate").**Theorem**: If there is one nondegenerate conic in a linear system, then there are at most three degenerate conics. However, there exist linear systems where all the conics are degenerate.

**Theorem**: If a real linear system of conics has finitely many base points, then there can be 0, 1, 2, 3, or 4 of them, but no more than four. However, there exist linear systems with infinitely many base points.**Theorem**: Every point in the plane which is not a base point lies on exactly one conic in the linear system.

The following examples list all the types of linear systems of conics,
up to "real projective equivalence" (real linear transformations of
the homogeneous coordinates [*x* : *y* : *z*]). It
turns out there are only finitely many types of linear systems;
roughly speaking, they can be categorized by the number of base points
and degenerate conics, but there are the usual issues of
"multiplicity," and whether the base points have real coordinates.

The numbering system for types I, Ia, ..., V follows H. Levy's book. The Roman numerals I, ..., VIII represent eight complex projective equivalence classes, some of which are further subdivided into real projective equivalence classes, for a total of 13 types.

Click on the picture to see an animation.

Type I
t_{1}^{ . }(x^{2}-y^{2}) + t_{2}^{ . }(x^{2}-1).
These conics meet at four real base points. There are three degenerate conics: - [
*t*_{1}:*t*_{2}]=[1:0] The intersecting lines*x*^{2}-*y*^{2}=0, - [
*t*_{1}:*t*_{2}]=[0:1] The vertical parallel lines*x*^{2}-1=0, - [
*t*_{1}:*t*_{2}]=[-1:1] The horizontal parallel lines*y*^{2}-1=0.
To see how I rendered the conic sections in the picture to the
left, Click
Here to see a "side view," showing the cones in |

**MR 0167882 29 #5147**H. Levy, Projective and related geometries. The Macmillan Co., New York; second printing 1967.**MR 1384308 (97a:14013)**A. Degtyarev, Quadratic transformations RP^{2}-> RP^{2}, in Topology of real algebraic varieties and related topics, 61--71, AMS Transl. Ser. 2,**173**, 1996.**MR 1673756 (99m:14012)**R. Miranda, Linear systems of plane curves, Notices Amer. Math. Soc. (2)**46**(1999), 192-201.

- Conics - The Geometry Center
- Famous Curves - St. Andrews

- F. Etayo, L. Gonzalez-Vega, and N. del Rio,
*A new approach to characterizing the relative position of two ellipses depending on one parameter*, Computer Aided Geometric Design (4)**23**(2006), 324-350. - C.-S. Chiang and H.-L. Liu,
*The tangency problem of variable radius circle to lines, circles and ellipses*, Ninth International Conference on Computer Aided Design and Computer Graphics, 2005. - Math Forum Internet Library
- Wikipedia entries on: Linear system of divisors, Degenerate conics.