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CoordinateTransformData
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CoordinateTransformData

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gives the value of the specified property for the coordinate transformation t.

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CoordinateTransformData[t,property,{x1,x2,…,xn}]

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 all

Basic Examples  (1)Summary of the most common use cases

Conversion between spherical and Cartesian coordinates in three dimensions:

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-w9z0mu
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0fq4f6mo4xjy6-uj1fi8
Out[2]=2

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:

In[1]:=1
✖
https://wolfram.com/xid/0fq4f6mo4xjy6-rvvamd
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-g0an8p
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0fq4f6mo4xjy6-y87a8x
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-bb5hwr
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0fq4f6mo4xjy6-lbqpxj
Out[2]=2

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:

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-l8518y
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-2iae3j
Out[1]=1

Properties  (1)

Get a list of possible properties:

In[1]:=1
✖
https://wolfram.com/xid/0fq4f6mo4xjy6-her1he
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0fq4f6mo4xjy6-wkrk71
Out[2]=2

Property Values  (2)

A property value can be any valid Wolfram Language expression:

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-qe6653
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0fq4f6mo4xjy6-49uco1
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0fq4f6mo4xjy6-bzyd8d
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-b3dhry
Out[1]=1

This function can be applied to points:

In[2]:=2
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https://wolfram.com/xid/0fq4f6mo4xjy6-jex9un
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0fq4f6mo4xjy6-63wxdo
Out[3]=3

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

In[4]:=4
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https://wolfram.com/xid/0fq4f6mo4xjy6-oci0j9
Out[4]=4

Detailed Properties  (4)

Convert a point in Cartesian coordinates to polar coordinates:

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-pi0n5a
Out[1]=1

Convert that point back to Cartesian coordinates:

In[2]:=2
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https://wolfram.com/xid/0fq4f6mo4xjy6-8qdcuj
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0fq4f6mo4xjy6-f4ad11
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-kmaymt
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0fq4f6mo4xjy6-cygkb6
Out[2]=2

The two matrices are inverses of each other:

In[3]:=3
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https://wolfram.com/xid/0fq4f6mo4xjy6-5rv2om
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-oftx70
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-544ei0
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0fq4f6mo4xjy6-zq6vjl
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0fq4f6mo4xjy6-pxg95l
Out[3]=3

Applications  (1)Sample problems that can be solved with this function

Look up the standard names of the variables of oblate spheroidal coordinates:

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-hrk67
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0fq4f6mo4xjy6-4uygap
Out[2]=2

Find the ranges of the different coordinates:

In[3]:=3
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https://wolfram.com/xid/0fq4f6mo4xjy6-n3qxm1
Out[3]=3

Plot a surface as and vary over their complete range:

In[4]:=4
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https://wolfram.com/xid/0fq4f6mo4xjy6-xmjd64
Out[4]=4

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

In[5]:=5
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https://wolfram.com/xid/0fq4f6mo4xjy6-x2588p
Out[5]=5

In the last plot, add a legend:

In[6]:=6
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https://wolfram.com/xid/0fq4f6mo4xjy6-p25mzu
Out[6]=6

Assemble all the pieces:

In[7]:=7
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https://wolfram.com/xid/0fq4f6mo4xjy6-l6nz5m
Out[7]=7

Properties & Relations  (2)Properties of the function, and connections to other functions

CoordinateTransformData[ent,"Mapping",pt] is effectively CoordinateTransform[ent,pt]:

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-eoipnr
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0fq4f6mo4xjy6-k3sv8x
Out[1]=1

The starting point is singular, with a degenerate metric:

In[2]:=2
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https://wolfram.com/xid/0fq4f6mo4xjy6-y857rf
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0fq4f6mo4xjy6-dtet27
Out[3]=3

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:

In[1]:=1
✖
https://wolfram.com/xid/0fq4f6mo4xjy6-rxmuf3
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0fq4f6mo4xjy6-kzbsae
Out[2]=2
Wolfram Research (2012), CoordinateTransformData, Wolfram Language function, https://reference.wolfram.com/language/ref/CoordinateTransformData.html.
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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.

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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

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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 ]}

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@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 ]}

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@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 ]}