This is a list of some of the most commonly used coordinate transformations.
2-dimensional
Let (''x'', ''y'') be the standard
Cartesian coordinates, and (''r'', ''θ'') the standard
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
.
To Cartesian coordinates
From polar coordinates
:
From log-polar coordinates
:
By using complex numbers
, the transformation can be written as
:
That is, it is given by the complex exponential function.
From bipolar coordinates
:
From 2-center bipolar coordinates
:
From Cesàro equation
:
To polar coordinates
From Cartesian coordinates
:
Note: solving for
returns the resultant angle in the first quadrant (
). To find
, one must refer to the original Cartesian coordinate, determine the quadrant in which
lies (for example, (3,−3)
artesianlies in QIV), then use the following to solve for
:
*For
in QI:
*:
*For
in QII:
*:
*For
in QIII:
*:
*For
in QIV:
*:
The value for
must be solved for in this manner because for all values of
,
is only defined for
, and is periodic (with period
). This means that the inverse function will only give values in the domain of the function, but restricted to a single period. Hence, the range of the inverse function is only half a full circle.
Note that one can also use
:
From 2-center bipolar coordinates
:
Where 2''c'' is the distance between the poles.
To log-polar coordinates from Cartesian coordinates
:
Arc-length and curvature
In Cartesian coordinates
:
In polar coordinates
:
3-dimensional
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
, with θ the angle measured away from the +Z axis (a
see conventions in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range of 180°, running from 0° to 180°, and does not pose any problem when calculated from an arccosine, but beware for an arctangent.
If, in the alternative definition, ''θ'' is chosen to run from −90° to +90°, in opposite direction of the earlier definition, it can be found uniquely from an arcsine, but beware of an arccotangent. In this case in all formulas below all arguments in ''θ'' should have sine and cosine exchanged, and as derivative also a plus and minus exchanged.
All divisions by zero result in special cases of being directions along one of the main axes and are in practice most easily solved by observation.
To Cartesian coordinates
From spherical coordinates
:
So for the volume element:
:
From cylindrical coordinates
:
So for the volume element:
:
To spherical coordinates
From Cartesian coordinates
:
See also the article on
atan2 for how to elegantly handle some edge cases.
So for the element:
:
From cylindrical coordinates
:
To cylindrical coordinates
From Cartesian coordinates
:
:
From spherical coordinates
:
Arc-length, curvature and torsion from Cartesian coordinates
:
See also
*
Geographic coordinate conversion
In geodesy, conversion among different geographic coordinate systems is made necessary by the different geographic coordinate systems in use across the world and over time. Coordinate conversion is composed of a number of different types of convers ...
*
Transformation matrix
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then
T( \mathbf x ) = A \mathbf x
for some m \times n matrix ...
References
*
{{DEFAULTSORT:Canonical Coordinate Transformations
Coordinate transformations
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 sign ...
Coordinate systems
Hamiltonian mechanics