In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and more specifically in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding
parametric equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
s of a
curve, a
surface, or, more generally, 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 a
variety, defined by an
implicit equation. The inverse process is called
implicitization
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ob ...
.
"To parameterize" by itself means "to express in terms 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".
Parametrization is a
mathematical process consisting of expressing the state of a
system
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and express ...
,
process or model as a
function of some independent quantities called
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. The state of the system is generally determined by a finite set of
coordinates, and the parametrization thus consists of one
function of several real variables for each coordinate. The number of parameters is the number of
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
of the system.
For example, the position of a
point that moves on a
curve in
three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
is determined by the time needed to reach the point when starting from a fixed origin. If are the coordinates of the point, the movement is thus described by a parametric equation
:
where is the parameter and denotes the time. Such a parametric equation completely determines the curve, without the need of any interpretation of as time, and is thus called a ''parametric equation'' of the curve (this is sometimes abbreviated by saying that one has a ''parametric curve''). One similarly gets the parametric equation of a surface by considering functions of two parameters and .
Non-uniqueness
Parametrizations are not generally
unique. The ordinary
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called '' parameters'') are required to determine the position of an element (i.e., point). This is the inform ...
object can be parametrized (or "coordinatized") equally efficiently with
Cartesian coordinates (''x'', ''y'', ''z''),
cylindrical polar coordinates (
ρ,
φ,
''z''),
spherical coordinates (
''r'', φ, θ) or other
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
s.
Similarly, the color space of human
trichromatic color vision can be parametrized in terms of the three colors red, green and blue,
RGB, or with cyan, magenta, yellow and black,
CMYK
The CMYK color model (also known as process color, or four color) is a subtractive color model, based on the CMY color model, used in color printing, and is also used to describe the printing process itself. The abbreviation ''CMYK'' refers ...
.
Dimensionality
Generally, the minimum number of parameters required to describe a model or geometric object is equal to its
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 coord ...
, and the scope of the parameters—within their allowed ranges—is the
parameter space The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for ...
. Though a good set of parameters permits identification of every point in the object space, it may be that, for a given parametrization, different parameter values can refer to the same point. Such mappings are
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
but not
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
. An example is the pair of cylindrical polar coordinates (ρ, φ, ''z'') and (ρ, φ + 2π, ''z'').
Invariance
As indicated above, there is arbitrariness in the choice of parameters of a given model, geometric object, etc. Often, one wishes to determine intrinsic properties of an object that do not depend on this arbitrariness, which are therefore independent of any particular choice of parameters. This is particularly the case in physics, wherein parametrization
invariance (or 'reparametrization invariance') is a guiding principle in the search for
physically acceptable theories (particularly in
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
).
For example, whilst the location of a fixed point on some curved line may be given by a set of numbers whose values depend on how the curve is parametrized, the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...
(appropriately defined) of the curve between ''two'' such fixed points will be independent of the particular choice of parametrization (in this case: the method by which an arbitrary point on the line is uniquely indexed). The length of the curve is therefore a parameterization-invariant quantity. In such cases parameterization is a mathematical tool employed to extract a result whose value does not depend on, or make reference to, the details of the parameterization. More generally, parametrization invariance of a physical theory implies that either 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 coord ...
ality or the volume of the parameter space is larger than is necessary to describe the physics (the quantities of physical significance) in question.
Though the theory of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
can be expressed without reference to a coordinate system, calculations of physical (i.e. observable) quantities such as the curvature of
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
invariably involve the introduction of a particular coordinate system in order to refer to spacetime points involved in the calculation. In the context of general relativity then, the choice of coordinate system may be regarded as a method of 'parameterizing' the spacetime, and the insensitivity of the result of a calculation of a physically-significant quantity to that choice can be regarded as an example of parameterization invariance.
As another example, physical theories whose observable quantities depend only on the ''relative'' distances (the ratio of distances) between pairs of objects are said to be
scale invariant. In such theories any reference in the course of a calculation to an absolute distance would imply the introduction of a parameter to which the theory is invariant.
Examples
*
Boy's surface
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. He discovered it on assignment from David Hilbert to prove that the projective plane ''could not'' be immersed in 3-space ...
*
McCullagh's parametrization of the Cauchy distributions
*
Parametrization (climate), the parametric representation of
general circulation model
A general circulation model (GCM) is a type of climate model. It employs a mathematical model of the general circulation of a planetary atmosphere or ocean. It uses the Navier–Stokes equations on a rotating sphere with thermodynamic ter ...
s and
numerical weather prediction
*
Singular isothermal sphere profile
*
Lambda-CDM model
The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parameterization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda ( Greek Λ) associated ...
, the standard
model of
Big Bang
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
cosmology
Techniques
*
Feynman parametrization
*
Schwinger parametrization
Schwinger parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops.
Using the well-known observation that
:\frac=\frac\int^\infty_0 du \, u^e^,
Julian Schwinger noticed that one may ...
*
Solid modeling
*
Dependency injection
References
{{reflist
External links
Brief Description of Parameterizationfrom
Oregon State University
Oregon State University (OSU) is a public land-grant, research university in Corvallis, Oregon. OSU offers more than 200 undergraduate-degree programs along with a variety of graduate and doctoral degrees. It has the 10th largest engineering ...
, and why it is useful, and a list of papers on the subject.
Coordinate systems