A locus (plural: loci) in mathematics is the collection of points that satisfy a stated condition or a combination of conditions. In elementary geometry a locus is often a curve in a plane or a surface in space, but it can be any subset of a space defined by rules. The concept helps translate geometric constraints into visual or analytic forms and is a common tool in problem solving and construction.
Definition and basic properties
Formally, a locus is a set of points P for which a property or predicate is true. The property might be a distance relation, an angular constraint, an algebraic equation, or a combination of inequalities. In analytic geometry loci are expressed by equations or systems of equations; in classical synthetic geometry they arise from constructions with compass and straightedge. A locus can be finite, infinite, continuous (a curve or surface) or discrete (a finite set of points).
Common examples
- Geometry: the perpendicular bisector of a segment is the locus of points equidistant from the segment's endpoints.
- Set-defined loci: given two fixed points, the set of points with constant sum of distances is an ellipse.
- The set of points at a fixed distance from a center is a circle in a plane.
- In three dimensions a plane section or a sphere is a typical surface locus.
- Some loci are curves such as parabolas and hyperbolas; these can be described by algebraic equations or focus-directrix conditions, related to curve theory.
History and methods
The study of loci goes back to classical Greek geometry where constructions identified sets of solutions to problems. With the development of analytic geometry by Descartes and others, loci became expressible by equations, allowing algebraic manipulation and classification. Modern treatments use set theory and topology to describe continuity, closure, and dimensionality of loci.
Applications and notable distinctions
Loci appear in geometric constructions, optimization, kinematics (paths traced by moving points), and complex analysis (level sets and argument loci). Important distinctions include the difference between a locus and a trajectory (a locus ignores time or order), and between an implicit locus defined by an equation and a parametric curve traced by a moving point. Degenerate loci—single points or intersecting lines—also occur when conditions are restrictive.
Understanding loci helps bridge visual intuition and algebraic representation, making them a fundamental concept in geometry, algebra, and applied mathematics.