Conic Section - Pencil of Conics

Pencil of Conics

A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.

In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points. Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.

A pencil of conics can represented algebraically in the following way. Let C₁ and C₂ be two distinct conics in a projective plane defined over an algebraically closed field K. For every pair λ, μ of elements of K, not both zero, the expression:

represents a conic in the pencil determined by C₁ and C₂. This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation defining the object.) Thinking of C₁, say, as a ternary quadratic form, then C₁ = 0 is the equation of the "conic C₁". Another concrete realization would be obtained by thinking of C₁ as the 3×3 symmetric matrix which represents it. If C₁ and C₂ have such concrete realizations then every member of the above pencil will as well. Since the setting uses homogeneous coordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic if they differ by a non-zero multiplicative constant.