General form with generators

There is a more powerful and unified way to describe a pencil: starting from two given lines, called generators. Combining them linearly yields a pencil that is proper or improper according to whether the two lines are incident or parallel. This section presents the two-generator form, the convenient single-parameter notation $r_1 + k\,r_2 = 0$ and the pitfall of the generator excluded from the parametrisation.

• The two-generator pencil
• The single-parameter notation