CoordinateTransformData
✖
CoordinateTransformData
gives the value of the specified property for the coordinate transformation t.
gives the value of the property evaluated at the point {x1,x2,…,xn}.
Details

- Transformations can be entered in the form oldchart->newchart, where oldchart and newchart are valid chart specifications available from CoordinateChartData.
- Coordinate transformation standard names are triples of the form {oldsys->newsys,metric,dim}, where {oldsys,metric,dim} and {newsys,metric,dim} are valid charts available from CoordinateChartData.
- CoordinateTransformData[] gives a list of available coordinate transformations, including only low-dimensional members of infinite families.
- CoordinateTransformData[t] is equivalent to CoordinateTransformData[t,"StandardName"].
- CoordinateTransformData["Properties"] returns a list of available properties.
- When no evaluation point {x1,x2,…,xn} is specified, properties are typically pure functions expecting a list of length n.
- Available properties include:
-
"Mapping" mapping from old to new coordinates "MappingJacobian" Jacobian matrix of the mapping "MappingJacobianDeterminant" determinant of the mapping Jacobian matrix "InverseMappingJacobian" inverse of the Jacobian matrix of the mapping "OrthonormalBasisRotation" rotation matrix between the orthonormal bases of the charts "StandardName" Wolfram Language standard name
Examples
open allclose allBasic Examples (1)Summary of the most common use cases
Scope (12)Survey of the scope of standard use cases
Names (5)
CoordinateTransformData[] returns a list of available coordinate transformations, including only low-dimensional members of infinite families:

https://wolfram.com/xid/0fq4f6mo4xjy6-rvvamd

Pairs {oldsys->newsys,dim} are equivalent to {oldsys->newsys,"Euclidean",dim}:

https://wolfram.com/xid/0fq4f6mo4xjy6-g0an8p


https://wolfram.com/xid/0fq4f6mo4xjy6-y87a8x

When an evaluation point is given, the dimension may be omitted from the first argument:

https://wolfram.com/xid/0fq4f6mo4xjy6-bb5hwr


https://wolfram.com/xid/0fq4f6mo4xjy6-lbqpxj

If one of the coordinate charts can be given to CoordinateChartData without a dimension specification, then the dimension specification can be omitted from CoordinateTransformData as well:

https://wolfram.com/xid/0fq4f6mo4xjy6-l8518y

A transform between two charts on the sphere of radius r:

https://wolfram.com/xid/0fq4f6mo4xjy6-2iae3j

Properties (1)
Property Values (2)
A property value can be any valid Wolfram Language expression:

https://wolfram.com/xid/0fq4f6mo4xjy6-qe6653


https://wolfram.com/xid/0fq4f6mo4xjy6-49uco1


https://wolfram.com/xid/0fq4f6mo4xjy6-bzyd8d

For most properties, if no evaluation point is given, then CoordinateTransformData will return a pure function:

https://wolfram.com/xid/0fq4f6mo4xjy6-b3dhry

This function can be applied to points:

https://wolfram.com/xid/0fq4f6mo4xjy6-jex9un


https://wolfram.com/xid/0fq4f6mo4xjy6-63wxdo

Certain descriptive properties, which manifestly do not depend on the variables, never return a pure function as a property value:

https://wolfram.com/xid/0fq4f6mo4xjy6-oci0j9

Detailed Properties (4)
Convert a point in Cartesian coordinates to polar coordinates:

https://wolfram.com/xid/0fq4f6mo4xjy6-pi0n5a

Convert that point back to Cartesian coordinates:

https://wolfram.com/xid/0fq4f6mo4xjy6-8qdcuj

Convert a generic point in polar coordinates to Cartesian coordinates, specifying the transform as a Rule of two charts:

https://wolfram.com/xid/0fq4f6mo4xjy6-f4ad11

Give the Jacobian of the mapping from polar to Cartesian coordinates:

https://wolfram.com/xid/0fq4f6mo4xjy6-kmaymt

Give the inverse of the Jacobian of the mapping from polar to Cartesian coordinates:

https://wolfram.com/xid/0fq4f6mo4xjy6-cygkb6

The two matrices are inverses of each other:

https://wolfram.com/xid/0fq4f6mo4xjy6-5rv2om

Compute the determinant of the mapping from spherical to Cartesian coordinates:

https://wolfram.com/xid/0fq4f6mo4xjy6-oftx70

The rotation matrix that rotates the spherical basis vectors into the Cartesian basis vectors
:

https://wolfram.com/xid/0fq4f6mo4xjy6-544ei0

When this matrix is applied to expressed in terms of
, the result is
:

https://wolfram.com/xid/0fq4f6mo4xjy6-zq6vjl

Similarly, the matrix applied to expressed in terms of
results in
:

https://wolfram.com/xid/0fq4f6mo4xjy6-pxg95l

Applications (1)Sample problems that can be solved with this function
Look up the standard names of the variables of oblate spheroidal coordinates:

https://wolfram.com/xid/0fq4f6mo4xjy6-hrk67

Use the transformation to Cartesian coordinates to visualize constant-coordinate surfaces:

https://wolfram.com/xid/0fq4f6mo4xjy6-4uygap

Find the ranges of the different coordinates:

https://wolfram.com/xid/0fq4f6mo4xjy6-n3qxm1

Plot a surface as
and
vary over their complete range:

https://wolfram.com/xid/0fq4f6mo4xjy6-xmjd64

Repeat the processing for the other coordinates, using None to get the correct sequence of colors:

https://wolfram.com/xid/0fq4f6mo4xjy6-x2588p

In the last plot, add a legend:

https://wolfram.com/xid/0fq4f6mo4xjy6-p25mzu


https://wolfram.com/xid/0fq4f6mo4xjy6-l6nz5m

Properties & Relations (2)Properties of the function, and connections to other functions
CoordinateTransformData[ent,"Mapping",pt] is effectively CoordinateTransform[ent,pt]:

https://wolfram.com/xid/0fq4f6mo4xjy6-eoipnr

CoordinateTransformData checks that inputs obey the coordinate range assumptions of charts:

https://wolfram.com/xid/0fq4f6mo4xjy6-k3sv8x


The starting point is singular, with a degenerate metric:

https://wolfram.com/xid/0fq4f6mo4xjy6-y857rf

Extract the general formula using the two-argument form and apply it to extend to singular points:

https://wolfram.com/xid/0fq4f6mo4xjy6-dtet27

Possible Issues (1)Common pitfalls and unexpected behavior
If one or both coordinate systems are given without parameters, and adding the default parameters would result in a duplicate parameter name, one of the default parameters will be renamed to prevent the clash:

https://wolfram.com/xid/0fq4f6mo4xjy6-rxmuf3


If both coordinate systems have user-specified parameter names, no renaming is done:

https://wolfram.com/xid/0fq4f6mo4xjy6-kzbsae

Wolfram Research (2012), CoordinateTransformData, Wolfram Language function, https://reference.wolfram.com/language/ref/CoordinateTransformData.html.
Text
Wolfram Research (2012), CoordinateTransformData, Wolfram Language function, https://reference.wolfram.com/language/ref/CoordinateTransformData.html.
Wolfram Research (2012), CoordinateTransformData, Wolfram Language function, https://reference.wolfram.com/language/ref/CoordinateTransformData.html.
CMS
Wolfram Language. 2012. "CoordinateTransformData." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CoordinateTransformData.html.
Wolfram Language. 2012. "CoordinateTransformData." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CoordinateTransformData.html.
APA
Wolfram Language. (2012). CoordinateTransformData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CoordinateTransformData.html
Wolfram Language. (2012). CoordinateTransformData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CoordinateTransformData.html
BibTeX
@misc{reference.wolfram_2025_coordinatetransformdata, author="Wolfram Research", title="{CoordinateTransformData}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/CoordinateTransformData.html}", note=[Accessed: 08-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_coordinatetransformdata, organization={Wolfram Research}, title={CoordinateTransformData}, year={2012}, url={https://reference.wolfram.com/language/ref/CoordinateTransformData.html}, note=[Accessed: 08-July-2025
]}