CoordinateTransform

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`CoordinateTransform`

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CoordinateTransform[t,pt]

performs the coordinate transformation t on the point pt.

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CoordinateTransform[t,{pt₁,pt₂,…}]

transforms several points.

Details

Transformations can be entered in the form oldchart->newchart, where oldchart and newchart are valid chart specifications available from CoordinateChartData.
Transformations can additionally be given as CoordinateTransformData standard names {oldsys->newsys,metric,dim}, where {oldsys,metric,dim} and {newsys,metric,dim} are valid charts available from CoordinateChartData. The short form in which dimension is omitted may be used.
CoordinateTransform[t,pt] is effectively equivalent to CoordinateTransformData[t,"Mapping",pt].
CoordinateTransform automatically threads over arrays of coordinate lists.

Examples

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Basic Examples (2)Summary of the most common use cases

Change a generic point in polar coordinates to Cartesian coordinates:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-imyf3d

Out[1]=1

Change a specific point in Cartesian coordinates to polar coordinates:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-2vf2hf

Out[1]=1

Scope (5)Survey of the scope of standard use cases

Give the {x,y,z} values for a point expressed in spherical coordinates:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-nzq3is

Out[1]=1

Change a point in prolate spheroidal to spherical coordinates, specifying a parameter for prolate spheroidal coordinates:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-w7bq0i

Out[1]=1

The same transformation, expressed more verbosely:

In[2]:=2

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https://wolfram.com/xid/0h2jrjkc5e-w3b32s

Out[2]=2

Transform coordinates on the sphere of radius r to corresponding values in the stereographic projection:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-4t0avr

Out[1]=1

Transform several points at once from cylindrical to Cartesian coordinates:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-mnihlj

Out[1]=1

Transform a matrix of points from Cartesian to spherical coordinates:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-k2x4i6

Out[1]=1

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

Convert a curve in non-Cartesian coordinates to a corresponding Cartesian expression for purposes of visualization:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-fsy8xd

In[2]:=2

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https://wolfram.com/xid/0h2jrjkc5e-2vdbnn

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0h2jrjkc5e-wj53gl

Out[3]=3

This curve is approximately 11.2 radii in length:

In[4]:=4

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https://wolfram.com/xid/0h2jrjkc5e-zcz0fh

Out[4]=4

Properties & Relations (8)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/0h2jrjkc5e-eoipnr

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-k3sv8x

Out[1]=1

This point violates the coordinate range assumption on the polar angle :

In[2]:=2

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https://wolfram.com/xid/0h2jrjkc5e-cfp8wt

Out[2]=2

Extract the symbolic transform from CoordinateTransformData to apply it to singular points:

In[3]:=3

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https://wolfram.com/xid/0h2jrjkc5e-dtet27

Out[3]=3

The reverse mapping is not well-defined at this point:

In[4]:=4

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https://wolfram.com/xid/0h2jrjkc5e-l144qv

Out[4]=4

CoordinateTransform preserves the shape of arrays:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-bv0cjq

Out[1]=1

This includes empty arrays:

In[2]:=2

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https://wolfram.com/xid/0h2jrjkc5e-jjfwrv

Out[2]=2

CoordinateTransform changes the coordinate values of points:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-8ymbgz

Out[1]=1

TransformedField changes the coordinate expressions for fields:

In[2]:=2

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https://wolfram.com/xid/0h2jrjkc5e-4naww

Out[2]=2

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-17wv03

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0h2jrjkc5e-yao322

Out[2]=2

ToPolarCoordinates is a special case of CoordinateTransform:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-74gc2b

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0h2jrjkc5e-6epzk4

Out[2]=2

FromSphericalCoordinates is a special case of CoordinateTransform:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-kt2cmc

Out[1]=1

ToSphericalCoordinates is a special case of CoordinateTransform:

In[1]:=1

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https://wolfram.com/xid/0h2jrjkc5e-ix6ikh

Out[1]=1

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