In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.^[1][2]

In other words, the set of the points that satisfy some property is often called the locus of a point satisfying this property. The use of singular in this formulation is a witness that, until the end of 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be located or may move.

Until the beginning of the 20th century, a geometrical shape (for example a curve) was not considered as an infinite set of points; rather, it was considered as an entity on which a point may be located or on which it moves. Thus a circle in the Euclidean plane was defined as the locus of a point that is at a given distance of a fixed point, the center of the circle. In modern mathematics, similar concepts are more frequently reformulated by describing shapes as sets; for instance, one says that the circle is the set of points that are at a given distance from the center.^[3]

In contrast to the set-theoretic view, the old formulation avoids considering infinite collections, as avoiding the actual infinite was an important philosophical position of earlier mathematicians.^[4][5]

Once set theory became the universal basis over which the whole mathematics is built,^[6] the term of locus became rather old-fashioned.^[7] Nevertheless, the word is still widely used, mainly for a concise formulation, for example:

More recently, techniques such as the theory of schemes, and the use of category theory instead of set theory to give a foundation to mathematics, have returned to notions more like the original definition of a locus as an object in itself rather than as a set of points.^[5]

Examples from plane geometry include:

Other examples of loci appear in various areas of mathematics. For example, in complex dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps.

To prove a geometric shape is the correct locus for a given set of conditions, one generally divides the proof into two stages:^[10]

We find the locus of the points P that have a given ratio of distances k = d1/d2 to two given points.

In this example we choose k = 3, A(−1, 0) and B(0, 2) as the fixed points.

A triangle ABC has a fixed side [AB] with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal.

We choose an orthonormal coordinate system such that A(−c/2, 0), B(c/2, 0). C(x, y) is the variable third vertex. The center of [BC] is M((2x + c)/4, y/2). The median from C has a slope y/x. The median AM has slope 2y/(2x + 3c).

The locus of the vertex C is a circle with center (−3c/4, 0) and radius 3c/4.

A locus can also be defined by two associated curves depending on one common parameter. If the parameter varies, the intersection points of the associated curves describe the locus.

A locus of points need not be one-dimensional (as a circle, line, etc.). For example,^[1] the locus of the inequality 2x + 3y – 6 < 0 is the portion of the plane that is below the line of equation 2x + 3y – 6 = 0.