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Locus (mathematics)

 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a locus (Latin
Latin

Latin is an Italic language, historically spoken in Latium and Ancient Rome. Through the Military history of the Roman Empire, Latin spread throughout the Mediterranean and a large part of Europe....
 for "place", plural loci) is a collection of points
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
 which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
. For example, in two-dimensional space a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 is the locus of points equidistant from two fixed points or from two parallel
Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more line s or plane , or a combination of these....
 lines.

conic sections may be defined in terms of loci:

Very complex geometric shapes may be described as the locus of zeros
Root (mathematics)

In mathematics, a root of a complex-valued Function is a member of the Domain of such that vanishes at , that is,In other words, a "root" of a function is a value for that produces a result of zero ....
 of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 or polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a locus (Latin
Latin

Latin is an Italic language, historically spoken in Latium and Ancient Rome. Through the Military history of the Roman Empire, Latin spread throughout the Mediterranean and a large part of Europe....
 for "place", plural loci) is a collection of points
Point (geometry)

In geometry, topology and related branches of mathematics a spatial point describes a specific object within a given space that consists of neither volume, area, length, nor any other higher dimensional analogue....
 which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
. For example, in two-dimensional space a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
 is the locus of points equidistant from two fixed points or from two parallel
Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more line s or plane , or a combination of these....
 lines.

Examples

Epitrochoid
The conic sections may be defined in terms of loci:
  • A circle
    Circle

    A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
     is the locus of points from which the distance
    Distance

    Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria ....
     to the center is a given value, the radius
    RADIUS

    Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
    .
  • An ellipse
    Ellipse

    In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
     is the locus of points, the sum of the distances from which to the foci
    Focus (geometry)

    In geometry, the foci, , are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola....
     is a given value.
  • A hyperbola
    Hyperbola

    In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....
     is the locus of points, the difference of the distances from which to the foci is a given value.
  • A parabola
    Parabola

    In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
     is the locus of points, the distances from which to the focus and to the directrix are equal.


Very complex geometric shapes may be described as the locus of zeros
Root (mathematics)

In mathematics, a root of a complex-valued Function is a member of the Domain of such that vanishes at , that is,In other words, a "root" of a function is a value for that produces a result of zero ....
 of a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 or polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
. Thus, for example, the quadric surfaces are defined as the loci of zeros of the quadratic polynomial
Quadratic polynomial

In mathematics, a quadratic polynomial or quadratic is a polynomial of degree of a polynomial two. A quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z....
s. More generally, the locus of zeros of a set of polynomials are known as an algebraic variety
Algebraic variety

In mathematics, an algebraic variety is essentially a set of points where a polynomial or set of polynomials attain a value of zero. Algebraic varieties are one of the central objects of study in classical algebraic geometry....
, the properties of which are studied in the branch of mathematics called algebraic geometry
Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry....
.

In complex dynamics
Complex dynamics

Complex dynamics the study of dynamical systems for which the phase space is a complex manifold. Complex analytic dynamics specifies more precisely that it is analytic functions whose dynamics it is to study....
 :
  • Bifurcation locus
    Bifurcation locus

    In complex dynamics, the Bifurcation theory 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....
  • Connectedness locus
    Connectedness locus

    In one-dimensional complex dynamics, the Connected space locus in a parameter space of polynomials or rational functions consists of those parameters for which the corresponding Julia set is connected....


Further examples of complex geometric shapes are generated by a point on a disk which is made to roll on a flat or curved surface.

See also

  • Focus (geometry)
    Focus (geometry)

    In geometry, the foci, , are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola....