See also: Frame of reference (homonymy)

In Mathematical, a frame of reference makes it possible to make correspond to each not of a space to NR Dimension S, a N-uplet of Scalaire S. In much of case, the scalars considered are real numbers, but it is possible to use complex numbers or elements of any body. More generally, the coordinates can come from a ring or another algebraic Structure related.

It is considered that space exists in itself independently of the choice of a particular frame of reference.

Examples

The case more the current is the concept of coordinates in geometry, to see the article Repérage in the plan and space : one chooses a benchmark called “origin”, and three “scales” of distinct directions which are not in the same plan (in the plan, two direction are enough). The coordinates of this point are called “X-coordinate”, “ordinate” and “dimension”, and are respectively noted X , there and Z . See also the analytical article Geometry .

In Geography, one associates a Longitude and a Latitude with geographical places; it is a frame of reference. In this case, parameterization is not single with the pole S Northern and Southern.

An example of frame of reference makes it possible to describe a point P in the Euclidean Espace $\backslash \; mathbb\; \{R\}\; ^n$ by a N - uplet:

$\backslash \; P\; =\; (r\_1,\dots ,\; r\_n)$

If a subset S of an Euclidean space is applied in way continuous E to another topological Espace, that defines the coordinates of the image of S . One can speak about parameterization of the image, since this process assigns numbers at the points. The correspondence is single only if the application is bijective.

Transformations

A transformation of coordinates is a conversion of a system to another to describe same space.

Certain choices of frame of reference can lead to Paradoxe S, for example in the vicinity of a Black hole, which can be solved by changing system. That is not however possible in true a mathematical Singularité.

Current systems

Some frames of reference usually used:

• the Cartesian Frame of reference used in a vector Space or a Espace refines of finished size.

• for all vector Space of finished size and all bases, the coefficients of the Vecteur S expressed in this base can be used like coordinates. To change basic is a transformation of coordinates, a linear Transformation which can be defined by a matrix.
• the curvilinear Frame of reference is a generalization, based on intersections of curves.
• the polar frames of reference:
• the cylindrical Frame of reference represents a point in space by a Angle, a distance to the origin and a height.
• the spherical Frame of reference represents a point in space by two angles and a distance to the origin. The geographical Frame of reference of it is derived.
• of the frames of reference generalized is used in Lagrangian Mécanique.

Systems used in astronomy

The Astronomie uses several frames of reference to note the direction of an celestial object:

• celestial frames of reference:

• horizontal Frame of reference, coordinated local related to a given point of the Ground;
• equatorial Frame of reference, definite starting from the terrestrial equatorial plan, and of the point γ (direction corresponding to the passage of the variation of the Sun of a negative value to a positive value);
• ecliptic Frame of reference, definite starting from the plan of revolution of the Earth around the Sun; this system can be geocentric or heliocentric (this last choice makes it possible to determine coordinates nonprone to the Précession);
• galactic Frame of reference, definite starting from a plan fundamental, selected once and for all, and located in the vicinity of the symmetry plane of our Galaxy (contained in the disc, or our Sun is); these coordinates are not prone to the secular precession, due to the displacement of the Solar system (approximately 250 km/s) within our Galaxy;
• extragalactic frames of reference:
• supergalactic Frame of reference, based on the plan of the Supercluster of Galaxy S local