Locus (mathematics)

In geometry, a locus is a collection of points that follow a specific rule. This rule is a condition that determines where each point can be located. A locus can take many forms, such as a single point, a line, a curve, or even a flat surface. The term comes from the Latin word for "place." Essentially, a locus describes the path or area where certain mathematical properties are satisfied.

To understand how a locus works, imagine a point moving across a plane. As it moves, it must constantly satisfy a given requirement. If the requirement is to stay exactly five centimeters from a center point, the point will trace a circle. If the requirement is to stay equal distances from two different points, the point will trace a perpendicular bisector. The shape is the physical result of the rule being applied to every possible location.

Mathematicians use different types of loci to define fundamental geometric shapes. For example, all conic sections are types of loci. A circle is the locus of points at a constant distance, called the radius, from a fixed center. A parabola is the locus of points equidistant from a fixed point, known as the focus, and a line called the directrix. An ellipse is defined by the sum of distances to two fixed foci remaining constant. A hyperbola is defined by the absolute difference between distances to two foci remaining constant.

There are also more complex ways to define a locus. One method involves using two associated curves that depend on a single common parameter. As that parameter changes, the curves move, and their intersection points trace out a new path. This intersection point describes the locus.

Another method uses ratios of distances. The circle of Apollonius is a specific locus. It is the set of points where the ratio of distances to two fixed points, such as A and B, remains a constant value, k.

The way mathematicians think about loci has changed over time. Before the beginning of the 20th century, mathematicians did not view curves as infinite sets of points. Instead, they viewed a shape as an entity where a point might be located or move. This was a philosophical choice to avoid the concept of the actual infinite. Once set theory became the foundation of modern mathematics, shapes were reformulated as sets. Today, we often say a circle is the set of all points at a specific distance from a center.

Even though the term "locus" can seem old-fashioned in a set-theoretic world, it remains very useful for concise descriptions. Mathematicians still use it to name specific sets in advanced study. A zero locus, or vanishing locus, is the set of points where a function takes the value of zero. A critical locus refers to the set of critical points of a differentiable function. A singular locus describes the singular points of an algebraic variety.

In more specialized fields, the concept of a locus is vital. In complex dynamics, the Mandelbrot set is characterized as a connectedness locus. This refers to a subset of the parameter set for a family of rational functions. Specifically, it is the set where the Julia set of the function is connected. Recently, new mathematical foundations like category theory and the theory of schemes have returned to viewing a locus as an object in itself, much like the original definitions used in the past.