In Year Two we saw that a line in the plane is determined by two geometric conditions: two points, or a point and the slope. But what happens if we impose just one condition on a line? The result is no longer a single line, but an entire family of lines: a pencil.
Learning to handle pencils is the key to solving elegantly dozens of analytic geometry problems: from finding the tangent to a curve from an external point, to determining a line that passes through a point and lies at a fixed distance from another, right up to finding lines that satisfy two conditions simultaneously. The chapter distinguishes proper pencils (lines through a fixed point) from improper pencils (lines parallel to a direction), introduces the powerful notation with two generators and shows the general method for determining the parameter that selects the required line.