Parametrization

# Parametrization

Discussion

Encyclopedia
Parametrization is the process of deciding and defining the parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

s necessary for a complete or relevant specification of a model
Scientific modelling
Scientific modelling is the process of generating abstract, conceptual, graphical and/or mathematical models. Science offers a growing collection of methods, techniques and theory about all kinds of specialized scientific modelling...

or geometric object.

Sometimes, this may only involve identifying certain parameters or variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

. If, for example, the model is of a wind turbine
Wind turbine
A wind turbine is a device that converts kinetic energy from the wind into mechanical energy. If the mechanical energy is used to produce electricity, the device may be called a wind generator or wind charger. If the mechanical energy is used to drive machinery, such as for grinding grain or...

with a particular interest in the efficiency of power generation, then the parameters of interest will probably include the number, length and pitch of the blades.

Most often, parametrization is a mathematical process involving the identification of a complete set of effective coordinates or degrees of freedom
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 independent pieces of information that go into the...

of the system
System
System is a set of interacting or interdependent components forming an integrated whole....

, process
Process (science)
In science, a process is every sequence of changes of a real object/body which is observable using the scientific method. Therefore, all sciences analyze and model processes....

or model, without regard to their utility in some design. Parametrization of a line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

, surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

or volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

, for example, implies identification of a set of coordinates that allows one to uniquely identify any point
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...

(on the line, surface, or volume) with an ordered list of numbers. Each of the coordinates can be defined parametrically in the form of a parametric curve (one-dimensional) or a parametric equation
Parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....

(2+ dimensions).

## Non-uniqueness

Parametrizations are not generally unique. The ordinary three-dimensional object can be parametrized (or 'coordinatized') equally efficiently with Cartesian coordinates (x,y,z), cylindrical polar coordinates (ρ, φ, z
Signed distance function
In mathematics and applications, the signed distance function of a set S in a metric space determines how close a given point x is to the boundary of S, with that function having positive values at points x inside S, it decreases in value as x approaches the boundary of S where the signed distance...

), spherical coordinates (r,φ,θ) or other coordinate systems.

Similarly, the color space of human trichromatic color vision can be parametrized in terms of the three colors red, green and blue, RGB, or alternatively with cyan, magenta and yellow, CMYK.

## Dimensionality

Generally, the minimum number of parameters required to describe a model or geometric object is equal to its dimension
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...

, and the scope of the parameters—within their allowed ranges—is the parameter space
Parameter space
In science, a parameter space is the set of values of parameters encountered in a particular mathematical model. Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function....

. Though a good set of parameters permits identification of every point in the parameter space, it may be that, for a given parametrization, different parameter values can refer to the same 'physical' point. Such mappings are surjective but not injective. An example is the pair of cylindrical polar coordinates (ρ,φ,z) and (ρ,φ + 2π,z).

## Parametrization 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
Invariant (physics)
In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.-Examples:In the current era, the immobility of polaris under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.Another...

(or 're-parametrization invariance') is a guiding principle in the search for physically acceptable theories (particularly in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It 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 different numbers depending on how the line is parametrized, the length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

of the line between two such fixed points will be independent of the choice of parametrization, even though it might have been computed using other coordinate systems.

Parametrization invariance implies that either the dimensionality or the volume of the parameter space is larger than that which is necessary to describe the physics in question (as exemplified in scale invariance
Scale invariance
In physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...

).

## Examples of parametrized models/objects

• Boy's surface
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...

• McCullagh's parametrization of the Cauchy distributions
• Parametrization (climate)
Parametrization (climate)
Parameterization in a weather or climate model within numerical weather prediction refers to the method of replacing processes that are too small-scale or complex to be physically represented in the model by a simplified process. This can be contrasted with other processes—e.g., large-scale flow of...

, the parametrical representation of general circulation models and numerical weather prediction
Numerical weather prediction
Numerical weather prediction uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in the 1950s that numerical weather predictions produced realistic...

• Singular isothermal sphere profile

## Parametrization techniques

• Feynman parametrization
Feynman parametrization
Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. However, it is sometimes useful in integration in areas of pure mathematics as well.Richard Feynman observed that:...

• Schwinger parametrization
• Solid modeling
Solid modeling
Solid modeling is a consistent set of principles for mathematical and computer modeling of three dimensional solids. Solid modeling is distinguished from related areas of Geometric modeling and Computer graphics by its emphasis on physical fidelity...

• Differential geometry of curves
Differential geometry of curves
Differential geometry of curves is the branch of geometry that dealswith smooth curves in the plane and in the Euclidean space by methods of differential and integral calculus....

• Estimand, the unknown parameter for which an estimation is sought
• Estimation theory
Estimation theory
Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the...

• Parametric surface
Parametric surface
A parametric surface is a surface in the Euclidean space R3 which is defined by a parametric equation with two parameters. Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence...

• Spline (mathematics)
Spline (mathematics)
In mathematics, a spline is a sufficiently smooth piecewise-polynomial function. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low-degree polynomials, while avoiding Runge's phenomenon for higher...

• Vector-valued function
Vector-valued function
A vector-valued function also referred to as a vector function is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector...