Locus

5.1 Locus

A locus is the set of every point in a plane that satisfies one fixed geometrical condition (or a fixed combination of conditions). Formally,

Here $P$ stands for a representative, general point of the set — we call $L$ the locus of $P$. A locus is fundamentally a set of points, and it can equally be pictured as the path traced by a point that moves while continuing to satisfy the condition (for instance, the path of a planet around the sun). The plural of locus is loci.

Illustrations, revisited algebraically as sets:

• The perpendicular bisector of $AB$ is $M = \{P \mid PA = PB\}$.
• The bisector of $\angle AOB$ is $D = \{P \mid P \text{ is equidistant from } OA \text{ and } OB\} = \{P \mid \angle POA = \angle POB\}$.
• The circle with centre $O$ and radius 4 is $L = \{P \mid OP = 4\}$. …