In many scientific fields, the degrees of freedom of a system is the number of

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s of the system that may vary independently. For example, a point in the plane has two degrees of freedom for

translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...

: its two

coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...

; a non-infinitesimal object on the plane might have additional degrees of freedoms related to its

orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...

. In

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, this notion is formalized as the

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

of a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

or an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...

. When ''degrees of freedom'' is used instead of ''dimension'', this usually means that the manifold or variety that models the system is only implicitly defined. See: *

Degrees of freedom (mechanics) In classical mechanics, physics, the number of degrees of freedom (DOF) of a mechanical system is the number of independent parameters required to completely specify its configuration or state. That number is an important property in the analysi ...

, number of independent motions that are allowed to the body or, in case of a

mechanism Mechanism may refer to: *Mechanism (economics), a set of rules for a game designed to achieve a certain outcome **Mechanism design, the study of such mechanisms *Mechanism (engineering), rigid bodies connected by joints in order to accomplish a ...

made of several bodies, number of possible independent relative motions between the pieces of the mechanism *

Degrees of freedom (physics and chemistry) In physics and chemistry, a degree of freedom is an independent physical parameter in the chosen parameterization of a physical system. More formally, given a parameterization of a physical system, the number of degrees of freedom is the smalles ...

, a term used in explaining dependence on parameters, or the dimensions of a phase space *

Degrees of freedom (statistics) In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. Estimates of statistical parameters can be based upon different amounts of information or data. The number of i ...

, the number of values in the final calculation of a statistic that are free to vary * Degrees of freedom problem, the problem of controlling motor movement given abundant degrees of freedom

See also