Locus (mathematics)

Overview

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a

**locus**(Latin

Latin

Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

for "place", plural

**) is a collection of points**

*loci*Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

which share a property. For example a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

may be defined as the locus of points in a plane at a fixed distance from a given point.

A locus may alternatively be described as the path through which a point moves to fulfill a given condition or conditions. So, for example, a circle may also be defined as the locus of a point moving so as to remain at a given distance from a fixed point.

The term occurs in complex dynamics

Complex dynamics

Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions.-Techniques:*General** Montel's theorem...

as:

- Bifurcation locusBifurcation locusIn complex dynamics, the bifurcation locus of a family of holomorphic functions informally is a locus of those maps for which the dynamical behavior changes drastically under a small perturbation of the parameter. Thus the bifurcation locus can be thought of as an analog of the Julia set in...
- Connectedness locusConnectedness locusIn one-dimensional complex dynamics, the connectedness locus in a parameter space of polynomials or rational functions consists of those parameters for which the corresponding Julia set is connected....

In general, a proof that a locus is a particular curve has two parts.

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