The pencil with two generators

There is a more powerful and unified way to describe a pencil: to build it from two given lines.

Property — Pencil with two generators

Let $r_1: a_1 x + b_1 y + c_1 = 0$ and $r_2: a_2 x + b_2 y + c_2 = 0$ be two distinct lines, written in implicit form. The equation $\alpha\,(a_1 x + b_1 y + c_1) + \beta\,(a_2 x + b_2 y + c_2) = 0, \qquad (\alpha;\beta)\ne(0;0),$ as the pair $(\alpha;\beta)$ varies, describes:

• a proper pencil if $r_1$ and $r_2$ are incident, with support their point of intersection;
• an improper pencil if $r_1$ and $r_2$ are parallel, with a common direction.

The lines $r_1$ and $r_2$ are called the generators of the pencil.

The strength of this notation is that it unifies the two types of pencil: one need only look at whether the generators meet or are parallel to know whether the pencil is proper or improper, without needing to compute the support explicitly.

Links

Topics: Pencils of lines
Concepts: Implicit equation · Pencil of lines · Improper pencil · Proper pencil · Generators · Support of the pencil
Functions: Line
Skills: Analytic geometry · Using formulae