# Computation of coordinate operations between two CRS

- Author:
Even Rouault

- Last Updated:
2023-08-26

## Introduction

When using **projinfo -s {crs_def} -t {crs_def}**,
**cs2cs {crs_def} {crs_def}** or the underlying
`proj_create_crs_to_crs()`

or `proj_create_operations()`

functions,
PROJ applies an algorithm to compute one or several candidate coordinate operations,
that can be expressed as a PROJ pipeline to transform between the source
and the target CRS. This document is about the description of this algorithm that
finds the actual operations to apply to be able later to perform transform coordinates.
So this is mostly about metadata management around coordinate operation methods,
and not about the actual mathematics used to implement those methods.
As a matter of fact with PROJ 6, there are about 60 000
lines of code dealing with "metadata" management (including conversions between PROJ
strings, all CRS WKT variants), to be compared to 30 000 for the purely computation part.

This document is meant as a plain text explanation of the code for developers, but also as a in-depth examination of what happens under the hood for curious PROJ users. It is important to keep in mind that it is not meant to be the ultimate source of truth of how coordinate operations should be computed. There are clearly implementation choices and compromises that can be questioned.

Let us start with an example to research operations between the NAD27 and NAD83 geographic CRS:

```
$ projinfo -s NAD27 -t NAD83 --summary --spatial-test intersects --grid-check none
Candidate operations found: 10
DERIVED_FROM(EPSG):1312, NAD27 to NAD83 (3), 1.0 m, Canada
DERIVED_FROM(EPSG):1313, NAD27 to NAD83 (4), 1.5 m, Canada - NAD27, at least one grid missing
DERIVED_FROM(EPSG):1241, NAD27 to NAD83 (1), 0.15 m, USA - CONUS including EEZ
DERIVED_FROM(EPSG):1243, NAD27 to NAD83 (2), 0.5 m, USA - Alaska including EEZ
DERIVED_FROM(EPSG):1573, NAD27 to NAD83 (6), 1.5 m, Canada - Quebec, at least one grid missing
EPSG:1462, NAD27 to NAD83 (5), 1.0 m, Canada - Quebec, at least one grid missing
EPSG:9111, NAD27 to NAD83 (9), 1.5 m, Canada - Saskatchewan, at least one grid missing
unknown id, Ballpark geographic offset from NAD27 to NAD83, unknown accuracy, World, has ballpark transformation
EPSG:8555, NAD27 to NAD83 (7), 0.15 m, USA - CONUS and GoM, at least one grid missing
EPSG:8549, NAD27 to NAD83 (8), 0.5 m, USA - Alaska, at least one grid missing
```

The algorithm involves many cases, so we will progress in the explanation from the most simple case to more complex ones. We document here the working of this algorithm as implemented in PROJ 8.0.0. The results of some examples might also be quite sensitive to the content of the PROJ database and the PROJ version used.

From a code point of view, the entry point of the algorithm is the C++
`osgeo::proj::operation::CoordinateOperation::createOperations()`

method.

It combines several strategies:

look up in the PROJ database for available operations

consider the pair (source CRS, target CRS) to synthetize operations depending on the nature of the source and target CRS.

## Geographic CRS to Geographic CRS, with known identifiers

With the above example of two geographic CRS, that have an identified identifier,
(projinfo internally resolves NAD27 to EPSG:4267 and NAD83 to EPSG:4269)
the algorithm will first search
in the coordinate operation related tables of the `proj.db`

if there are records
that list direct transformations between the source and the target CRS. The
transformations typically involve Helmert-style operations or datum shift based on
grids (more esoteric operations are possible).

A request similar to the following will be emitted:

```
$ sqlite3 proj.db "SELECT auth_name, code, name, method_name, accuracy FROM \
coordinate_operation_view WHERE \
source_crs_auth_name = 'EPSG' AND \
source_crs_code = '4267' AND \
target_crs_auth_name = 'EPSG' AND \
target_crs_code = '4269'"
EPSG|1241|NAD27 to NAD83 (1)|NADCON|0.15
EPSG|1243|NAD27 to NAD83 (2)|NADCON|0.5
EPSG|1312|NAD27 to NAD83 (3)|NTv1|1.0
EPSG|1313|NAD27 to NAD83 (4)|NTv2|1.5
EPSG|1462|NAD27 to NAD83 (5)|NTv1|1.0
EPSG|1573|NAD27 to NAD83 (6)|NTv2|1.5
EPSG|8549|NAD27 to NAD83 (8)|NADCON5 (2D)|0.5
EPSG|8555|NAD27 to NAD83 (7)|NADCON5 (2D)|0.15
EPSG|9111|NAD27 to NAD83 (9)|NTv2|1.5
ESRI|108003|NAD_1927_To_NAD_1983_PR_VI|NTv2|0.05
```

As we have found direct transformations, we will not attempt any more complicated research. One can note in the above result set that a ESRI:108003 operation was found, but as the source and target CRS are in the EPSG registry, and there are operations between those CRS in the EPSG registry itself, transformations from other authorities will be ignored (except if they are in the PROJ authority, which can be used as an override).

As those results all involve operations that does not have a perfect accuracy and that does not cover the area of use of the 2 CRSs, a 'Ballpark geographic offset from NAD27 to NAD83' operation is synthetized by PROJ (see Ballpark transformation)

## Filtering and sorting of coordinate operations

The last step is to filter and sort results in order of relevance.

The filtering takes into account the following criteria to decide which operations must be retained or discarded:

a minimum accuracy that the user might have expressed,

an area of use on which the coordinate operation(s) must apply

if the absence of grids needed by an operation must result in discarding it.

The sorting algorithm determines the order of relevance of the operations we got.
A comparison function compares pair of operations to determine which of the
two is the most relevant. This is implemented by the `operator ()`

method of the SortFunction structure.
When comparing two operations, the following criteria are used. The tests are
performed in the order they are listed below:

consider as more relevant an operation that can be expressed as a PROJ operation string (the database might list operations whose method is not (yet) implemented by PROJ)

if both operations evaluate identically with respect to the above criterion, consider as more relevant an operation that does not include a synthetic ballpark vertical transformation (occurs when there is a geoid model).

if both operations evaluate identically with respect to the above criterion, consider as more relevant an operation that does not include a synthetic ballpark horizontal transformation.

consider as more relevant an operation that refers to shift grids that are locally available.

consider as more relevant an operation that refers to grids that are available in one of the proj-datumgrid packages, but not necessarily locally available

consider as more relevant an operation that has a known accuracy.

if two operations have unknown accuracy, consider as more relevant an operation that uses grid(s) if the other one does not (grid based operations are assumed to be more precise than operations relying on a few parameters)

consider as more relevant an operation whose area of use is larger (note: the computation of the are of use is approximate, based on a bounding box)

consider as more relevant an operation that has a better accuracy.

in case of same accuracy, consider as more relevant an operation that does not use grids (operations that use only parameters will be faster)

consider as more relevant an operation that involves less transformation steps (transformation steps considered are the ones listed in the WKT output, not PROJ pipeline steps)

and for completeness, if two operations are comparable given all the above criteria, consider as more relevant the one which has the shorter name, and if they have the same length, consider as more relevant the one whose name comes last in lexicographic order (e.g. "FOO to BAR (3)" will have higher precedence than "FOO to BAR (2)")

Note

`proj_trans()`

, on the results returned by `proj_create_crs_to_crs()`

,
will not necessarily use the operation that
is listed in first position due to the above algorithm. `proj_trans()`

has more context, since it has the coordinate to transform, so it can compare
this coordinate to the area of use of operations. Typically, the above criteria
will favor an operation that has a larger area of use over another one with a
smaller area, due to it being more generally applicable. But once coordinates are known,
`proj_trans()`

can select an operation with a smaller
area of use that applies to the coordinate to transform.

## Geodetic/geographic CRS to Geodetic/geographic CRS, without known identifiers

In a number of situations, the source and/or target CRS do not have an identifier
(WKT without identifier, PROJ string, ..)
The first step is to try to find in the `proj.db`

a CRS of the same nature of
the CRS to identify and whose name exactly matches the one provided to the
`createOperations()`

method. If there is exactly one match and that the CRS are
"computationally" equivalent, then use the code of the CRS for further computations.

If this search did not succeed, or if the previous case with known CRS identifiers did not result in matches in the database, the search will be based on the datums. That is, a list of geographic CRS whose datum matches the datum of the source and target CRS is searched for in the database (by querying the geodetic_crs database table). If the datum has a known identifier, we will use it, otherwise we will look for an equivalent datum in the database based on the datum name.

Let's consider the case where the datum of the source CRS is EPSG:6171 "Reseau Geodesique Francais 1993" and the datum of the target CRS is EPSG:6258 "European Terrestrial Reference System 1989". For EPSG:6171, there are 10 matching (non-deprecated) geodetic CRSs:

EPSG:4171, RGF93, geographic 2D

EPSG:4964, RGF93, geocentric

EPSG:4965, RGF93, geographic 3D

EPSG:7042, RGF93 (lon-lat), geographic 3D

EPSG:7084, RGF93 (lon-lat), geographic 2D

IGNF:RGF93, RGF93 cartesiennes geocentriques, geocentric

IGNF:RGF93GDD, RGF93 geographiques (dd),geographic 2D

IGNF:RGF93GEODD, RGF93 geographiques (dd), geographic 3D

IGNF:RGF93G, RGF93 geographiques (dms), geographic 2D

IGNF:RGF93GEO, RGF93 geographiques (dms), geographic 3D

The first three entries from the EPSG dataset are typical: for each datum, one can define a geographic 2D CRS (latitude, longitude), a geographic 3D crs (latitude, longitude, ellipsoidal height) and a geocentric one. For that particular case, the EPSG dataset has also included two extra definitions corresponding to a longitude, latitude, [ellipsoidal height] coordinate system, as found in the official French IGNF registry. This IGNF registry has also definitions for a geographic 2D CRS (with an extra subtlety with an entry using decimal degree as unit and another one degree-minute-second), geographic 3D and geocentric.

For EPSG:6258, there are 7 matching (non-deprecated) geodetic CRSs:

EPSG:4258, ETRS89, geographic 2D

EPSG:4936, ETRS89, geocentric

EPSG:4937, ETRS89, geographic 3D

IGNF:ETRS89, ETRS89 cartesiennes geocentriques, geocentric

IGNF:ETRS89G, ETRS89 geographiques (dms), geographic 2D

IGNF:ETRS89GEO, ETRS89 geographiques (dms), geographic 3D

ESRI:104129, GCS_EUREF_FIN, geographic 2D

So the 3 typical EPSG entries, 3 equivalent (with long, lat ordering for the geographic CRS) and one from the ESRI registry;

PROJ can now test 10 x 7 different combinations of source x target CRSs, using the database searching method explained in the previous section. As soon as one of this combination returns at least one non-ballpark combination, the result set coming from that combination is used. PROJ will then add before that transformation a conversion between the source CRS and the first intermediate CRS, and will add at the end a conversion between the second intermediate CRS and the target CRS. Those conversions are conversion between geographic 2D and geographic 3D CRS or geographic 2D/3D and geocentric CRS.

This is done by the `createOperationsWithDatumPivot()`

method.

So if transforming between EPSG:7042, RGF93 (lon-lat), geographic 3D and EPSG:4936, ETRS89, geocentric, one get the following concatenated operation, chaining an axis order change, the null geocentric translation between RGF93 and ETRS89 (EPSG:1591), and a conversion between geographic and geocentric coordinates. This concatenated operation is assumed to have a perfect accuracy as both the initial and final operations are conversions, and the middle transformation accounts for the fact that the RGF93 datum is one realization of ETRS89, so they are equivalent for most purposes.

```
$ projinfo -s EPSG:7042 -t EPSG:4936
Candidate operations found: 1
-------------------------------------
Operation No. 1:
unknown id, axis order change (geographic3D horizontal) + RGF93 to ETRS89 (1) + Conversion from ETRS89 (geog2D) to ETRS89 (geocentric), 0 m, France
PROJ string:
+proj=pipeline +step +proj=unitconvert +xy_in=deg +xy_out=rad +step +proj=cart +ellps=GRS80
WKT2:2019 string:
CONCATENATEDOPERATION["axis order change (geographic3D horizontal) + RGF93 to ETRS89 (1) + Conversion from ETRS89 (geog2D) to ETRS89 (geocentric)",
SOURCECRS[
GEOGCRS["RGF93 (lon-lat)",
[...]
ID["EPSG",7042]]],
TARGETCRS[
GEODCRS["ETRS89",
[...]
ID["EPSG",4936]]],
STEP[
CONVERSION["axis order change (geographic3D horizontal)",
METHOD["Axis Order Reversal (Geographic3D horizontal)",
ID["EPSG",9844]],
ID["EPSG",15499]]],
STEP[
COORDINATEOPERATION["RGF93 to ETRS89 (1)",
[...]
METHOD["Geocentric translations (geog2D domain)",
ID["EPSG",9603]],
PARAMETER["X-axis translation",0,
LENGTHUNIT["metre",1],
ID["EPSG",8605]],
PARAMETER["Y-axis translation",0,
LENGTHUNIT["metre",1],
ID["EPSG",8606]],
PARAMETER["Z-axis translation",0,
LENGTHUNIT["metre",1],
ID["EPSG",8607]],
OPERATIONACCURACY[0.0],
ID["EPSG",1591],
REMARK["May be taken as approximate transformation RGF93 to WGS 84 - see code 1671."]]],
STEP[
CONVERSION["Conversion from ETRS89 (geog2D) to ETRS89 (geocentric)",
METHOD["Geographic/geocentric conversions",
ID["EPSG",9602]]]],
USAGE[
SCOPE["unknown"],
AREA["France"],
BBOX[41.15,-9.86,51.56,10.38]]]
```

## Geodetic/geographic CRS to Geodetic/geographic CRS, without direct transformation

Still considering transformations between geodetic/geographic CRS, but let's consider that the lookup in the database for a transformation between the source and target CRS (possibly going through the "equivalent" CRS based on the same datum as detailed in the previous section) leads to an empty set.

Of course, as most operations are invertible, one first tries to do a lookup switching the source and target CRS, and inverting the resulting operation(s):

```
$ projinfo -s NAD83 -t NAD27 --spatial-test intersects --summary
Candidate operations found: 10
INVERSE(DERIVED_FROM(EPSG)):1312, Inverse of NAD27 to NAD83 (3), 2.0 m, Canada
INVERSE(DERIVED_FROM(EPSG)):1313, Inverse of NAD27 to NAD83 (4), 1.5 m, Canada - NAD27
INVERSE(DERIVED_FROM(EPSG)):1241, Inverse of NAD27 to NAD83 (1), 0.15 m, USA - CONUS including EEZ
INVERSE(DERIVED_FROM(EPSG)):1243, Inverse of NAD27 to NAD83 (2), 0.5 m, USA - Alaska including EEZ
INVERSE(DERIVED_FROM(EPSG)):1573, Inverse of NAD27 to NAD83 (6), 1.5 m, Canada - Quebec, at least one grid missing
INVERSE(EPSG):1462, Inverse of NAD27 to NAD83 (5), 2.0 m, Canada - Quebec, at least one grid missing
INVERSE(EPSG):9111, Inverse of NAD27 to NAD83 (9), 1.5 m, Canada - Saskatchewan, at least one grid missing
unknown id, Ballpark geographic offset from NAD83 to NAD27, unknown accuracy, World, has ballpark transformation
INVERSE(EPSG):8555, Inverse of NAD27 to NAD83 (7), 0.15 m, USA - CONUS and GoM, at least one grid missing
INVERSE(EPSG):8549, Inverse of NAD27 to NAD83 (8), 0.5 m, USA - Alaska, at least one grid missing
```

That was an easy case. Now let's consider the transformation between the Australian CRS AGD84 and GDA2020. There is no direct transformation from AGD84 to GDA2020, or in the reverse direction, even when considering alternative geodetic CRS based on the underlying datums. PROJ will then do a cross-join in the coordinate_operation_view table to find the tuples (op1, op2) of coordinate operations such that:

SOURCE_CRS = op1.source_crs AND op1.target_crs = op2.source_crs AND op2.target_crs = TARGET_CRS or

SOURCE_CRS = op1.source_crs AND op1.target_crs = op2.target_crs AND op2.source_crs = TARGET_CRS or

SOURCE_CRS = op1.target_crs AND op1.source_crs = op2.source_crs AND op2.target_crs = TARGET_CRS or

SOURCE_CRS = op1.target_crs AND op1.source_crs = op2.target_crs AND op2.source_crs = TARGET_CRS

Depending on which case is selected, op1 and op2 should be reversed, before being concatenated.

This logic is implement by the `findsOpsInRegistryWithIntermediate()`

method.

Assuming that the proj-datumgrid-oceania package is installed, we get the following results for the AGD84 to GDA2020 coordinate operations lookup:

```
$ projinfo -s AGD84 -t GDA2020 --spatial-test intersects -o PROJ
Candidate operations found: 4
-------------------------------------
Operation No. 1:
unknown id, AGD84 to GDA94 (5) + GDA94 to GDA2020 (1), 0.11 m, Australia - AGD84
PROJ string:
+proj=pipeline +step +proj=axisswap +order=2,1 \
+step +proj=unitconvert +xy_in=deg +xy_out=rad \
+step +proj=hgridshift +grids=National_84_02_07_01.gsb \
+step +proj=push +v_3 \
+step +proj=cart +ellps=GRS80 \
+step +proj=helmert +x=0.06155 +y=-0.01087 +z=-0.04019 \
+rx=-0.0394924 +ry=-0.0327221 +rz=-0.0328979 \
+s=-0.009994 +convention=coordinate_frame \
+step +inv +proj=cart +ellps=GRS80 \
+step +proj=pop +v_3 \
+step +proj=unitconvert +xy_in=rad +xy_out=deg \
+step +proj=axisswap +order=2,1
-------------------------------------
Operation No. 2:
unknown id, AGD84 to GDA94 (2) + GDA94 to GDA2020 (1), 1.01 m, Australia - AGD84
PROJ string:
+proj=pipeline +step +proj=axisswap +order=2,1 \
+step +proj=unitconvert +xy_in=deg +xy_out=rad \
+step +proj=push +v_3 \
+step +proj=cart +ellps=aust_SA \
+step +proj=helmert +x=-117.763 +y=-51.51 +z=139.061 \
+rx=-0.292 +ry=-0.443 +rz=-0.277 +s=-0.191 \
+convention=coordinate_frame \
+step +proj=helmert +x=0.06155 +y=-0.01087 +z=-0.04019 \
+rx=-0.0394924 +ry=-0.0327221 +rz=-0.0328979 \
+s=-0.009994 +convention=coordinate_frame \
+step +inv +proj=cart +ellps=GRS80 \
+step +proj=pop +v_3 \
+step +proj=unitconvert +xy_in=rad +xy_out=deg \
+step +proj=axisswap +order=2,1
-------------------------------------
Operation No. 3:
unknown id, AGD84 to GDA94 (5) + GDA94 to GDA2020 (2), 0.15 m, unknown domain of validity
PROJ string:
+proj=pipeline +step +proj=axisswap +order=2,1 \
+step +proj=unitconvert +xy_in=deg +xy_out=rad \
+step +proj=hgridshift +grids=National_84_02_07_01.gsb \
+step +proj=hgridshift +grids=GDA94_GDA2020_conformal_and_distortion.gsb \
+step +proj=unitconvert +xy_in=rad +xy_out=deg \
+step +proj=axisswap +order=2,1
-------------------------------------
Operation No. 4:
unknown id, AGD84 to GDA94 (5) + GDA94 to GDA2020 (3), 0.15 m, unknown domain of validity
PROJ string:
+proj=pipeline +step +proj=axisswap +order=2,1 \
+step +proj=unitconvert +xy_in=deg +xy_out=rad \
+step +proj=hgridshift +grids=National_84_02_07_01.gsb \
+step +proj=hgridshift +grids=GDA94_GDA2020_conformal.gsb \
+step +proj=unitconvert +xy_in=rad +xy_out=deg \
+step +proj=axisswap +order=2,1
```

One can see that the selected intermediate CRS that has been used is GDA94. This is a completely novel behavior of PROJ 6 as opposed to the logic of PROJ.4 where datum transformations implied using EPSG:4326 / WGS 84 has the mandatory datum hub. PROJ 6 no longer hardcodes it as the mandatory datum hub, and relies on the database to find the appropriate hub(s). Actually, WGS 84 has been considered during the above lookup, because there are transformations between AGD84 and WGS 84 and WGS 84 and GDA2020. However those have been discarded in a step which we did not mention previously: just after the initial filtering of results and their sorting, there is a final filtering that is done. In the list of sorted results, given two operations A and B that have the same area of use, if B has an accuracy lower than A, and A does not use grids, or all the needed grids are available, then B is discarded.

If one forces the datum hub to be considered to be EPSG:4326, ones gets:

```
$ projinfo -s AGD84 -t GDA2020 --spatial-test intersects --pivot-crs EPSG:4326 -o PROJ
Candidate operations found: 2
-------------------------------------
Operation No. 1:
unknown id, AGD84 to WGS 84 (7) + Inverse of GDA2020 to WGS 84 (2), 4 m, Australia - AGD84
PROJ string:
+proj=pipeline +step +proj=axisswap +order=2,1 \
+step +proj=unitconvert +xy_in=deg +xy_out=rad \
+step +proj=push +v_3 \
+step +proj=cart +ellps=aust_SA \
+step +proj=helmert +x=-117.763 +y=-51.51 +z=139.061 \
+rx=-0.292 +ry=-0.443 +rz=-0.277 \
+s=-0.191 +convention=coordinate_frame \
+step +inv +proj=cart +ellps=GRS80 \
+step +proj=pop +v_3 \
+step +proj=unitconvert +xy_in=rad +xy_out=deg \
+step +proj=axisswap +order=2,1
-------------------------------------
Operation No. 2:
unknown id, AGD84 to WGS 84 (9) + Inverse of GDA2020 to WGS 84 (2), 4 m, Australia - AGD84
PROJ string:
+proj=pipeline +step +proj=axisswap +order=2,1 \
+step +proj=unitconvert +xy_in=deg +xy_out=rad \
+step +proj=hgridshift +grids=National_84_02_07_01.gsb \
+step +proj=unitconvert +xy_in=rad +xy_out=deg \
+step +proj=axisswap +order=2,1
```

Those operations are less accurate, since WGS 84 is assumed to be equivalent to GDA2020 with an accuracy of 4 metre. This is an instance demonstrating that using WGS 84 as a hub systematically can be sub-optimal.

There are still situations where the attempt to find a hub CRS does not work, because there is no such hub. This can occur for example when transforming from GDA94 to the latest realization at time of writing of WGS 84, WGS 84 (G1762). There are transformations between WGS 84 (G1762). Using the above described techniques, we would only find one non-ballpark operation taking the route: 1. Conversion from GDA94 (geog2D) to GDA94 (geocentric): synthetized by PROJ 2. Inverse of ITRF2008 to GDA94 (1): from EPSG 3. Inverse of WGS 84 (G1762) to ITRF2008 (1): from EPSG 4. Conversion from WGS 84 (G1762) (geocentric) to WGS 84 (G1762): synthetized by PROJ

This is not bad, but the global validity area of use is "Australia - onshore and EEZ", whereas GDA94 has a larger area of use. There is another road that can be taken by going through GDA2020 instead of ITRF2008. The GDA94 to GDA2020 transformations operate on the respective geographic CRS, whereas GDA2020 to WGS 84 (G1762) operate on the geocentric CRS. Consequently, GDA2020 cannot be identifier as a hub by a "simple" self-join SQL request on the coordinate operation table. This requires to do the join based on the datum referenced by the source and target CRS of each operation rather than the source and target CRS themselves. When there is a match, PROJ inserts the required conversions between geographic and geocentric CRS to have a consistent concatenated operation, like the following: 1. GDA94 to GDA2020 (1): from EPSG 2. Conversion from GDA2020 (geog2D) to GDA2020 (geocentric): synthetized by PROJ 3. GDA2020 to WGS 84 (G1762) (1): from EPSG 4. Conversion from WGS 84 (G1762) (geocentric) to WGS 84 (G1762) (geog2D): synthetized by PROJ

## Projected CRS to any target CRS

This actually extends to any Derived CRS, whose Projected CRS is a well-known particular case. Such transformations are done in 2 steps:

Use the inverse conversion of the derived CRS to its base CRS, typically an inverse map projection.

Find transformations from this base CRS to the target CRS. If the base CRS is the target CRS, this step can be skipped.

```
$ projinfo -s EPSG:32631 -t RGF93
Candidate operations found: 1
-------------------------------------
Operation No. 1:
unknown id, Inverse of UTM zone 31N + Inverse of RGF93 to WGS 84 (1), 1 m, France
PROJ string:
+proj=pipeline +step +inv +proj=utm +zone=31 +ellps=WGS84 +step +proj=unitconvert +xy_in=rad +xy_out=deg +step +proj=axisswap +order=2,1
```

This is implemented by the `createOperationsDerivedTo`

method

For the symmetric case, source CRS to a derived CRS, the above algorithm is applied by switching the source and target CRS, and then inverting the resulting operation(s). This is mostly a matter of avoiding to write very similar code twice. This logic is also applied to all below cases when considering the transformation between 2 different types of objects.

## Vertical CRS to a Geographic CRS

Such transformation is normally not meant as being used as standalone by PROJ users, but as an internal computation step of a Compound CRS to a target CRS.

In cases where we are lucky, the PROJ database will have a transformation registered between those:

```
$ projinfo -s "NAVD88 height" -t "NAD83(2011)" -o PROJ --spatial-test intersects
Candidate operations found: 11
-------------------------------------
Operation No. 1:
INVERSE(DERIVED_FROM(EPSG)):9229, Inverse of NAD83(2011) to NAVD88 height (3), 0.015 m, USA - CONUS - onshore
PROJ string:
+proj=vgridshift +grids=g2018u0.gtx +multiplier=1
```

But in cases where there is no match, the `createOperationsVertToGeog`

method
will be used to synthetize a ballpark vertical transformation, just taking care
of unit changes, and axis reversal in case the vertical CRS was a depth rather than
a height. Of course the results of such an operation are questionable, hence the
ballpark qualifier and a unknown accuracy advertized for such an operation.

## Vertical CRS to a Vertical CRS

Overall logic is similar to the above case. There might be direct operations in the PROJ database, involving grid transformations or simple offsets. The fallback case is to synthetize a ballpark transformation.

This is implemented by the `createOperationsVertToVert`

method

```
$ projinfo -s "NGVD29 depth (ftUS)" -t "NAVD88 height" --spatial-test intersects -o PROJ
Candidate operations found: 3
-------------------------------------
Operation No. 1:
unknown id, Inverse of NGVD29 height (ftUS) to NGVD29 depth (ftUS) + NGVD29 height (ftUS) to NGVD29 height (m) + NGVD29 height (m) to NAVD88 height (3), 0.02 m, USA - CONUS east of 89°W - onshore
PROJ string:
+proj=pipeline +step +proj=axisswap +order=1,2,-3 +step +proj=unitconvert +z_in=us-ft +z_out=m +step +proj=vgridshift +grids=vertcone.gtx +multiplier=0.001
-------------------------------------
Operation No. 2:
unknown id, Inverse of NGVD29 height (ftUS) to NGVD29 depth (ftUS) + NGVD29 height (ftUS) to NGVD29 height (m) + NGVD29 height (m) to NAVD88 height (2), 0.02 m, USA - CONUS 89°W-107°W - onshore
PROJ string:
+proj=pipeline +step +proj=axisswap +order=1,2,-3 +step +proj=unitconvert +z_in=us-ft +z_out=m +step +proj=vgridshift +grids=vertconc.gtx +multiplier=0.001
-------------------------------------
Operation No. 3:
unknown id, Inverse of NGVD29 height (ftUS) to NGVD29 depth (ftUS) + NGVD29 height (ftUS) to NGVD29 height (m) + NGVD29 height (m) to NAVD88 height (1), 0.02 m, USA - CONUS west of 107°W - onshore
PROJ string:
+proj=pipeline +step +proj=axisswap +order=1,2,-3 +step +proj=unitconvert +z_in=us-ft +z_out=m +step +proj=vgridshift +grids=vertconw.gtx +multiplier=0.001
```

## Compound CRS to a Geographic CRS

A typical example of a Compound CRS is a CRS made of a geographic or projected CRS as the horizontal component, and a vertical CRS. E.g. "NAD83 + NAVD88 height"

When the horizontal component of the compound source CRS is a projected CRS, we first look for the operation from this source CRS to another compound CRS made of the geographic CRS base of the projected CRS, like "NAD83 / California zone 1 (ftUS) + NAVD88 height" to "NAD83 + NAVD88 height", which ultimately goes to one of the above described case. Then we can consider the transformation from a compound CRS made of a geographic CRS to another geographic CRS.

It first starts by the vertical transformations from the vertical CRS of the source compound CRS to the target geographic CRS, using the strategy detailed in Vertical CRS to a Geographic CRS

What we did not mention is that when there is not a transformation registered between the vertical CRS and the target geographic CRS, PROJ attempts to find transformations between that vertical CRS and any other geographic CRS. This is clearly an approximation. If the research of the vertical CRS to the target geographic CRS resulted in operations that use grids that are not available, as another approximation, we research operations from the vertical CRS to the source geographic CRS for the vertical component.

Once we got those more or less accurate vertical transformations, we must consider the horizontal transformation(s). The algorithm iterates over all found vertical transformations and look for their target geographic CRS. This will be used as the interpolation CRS for horizontal transformations. PROJ will then look for available transformations from the source geographic CRS to the interpolation CRS and from the interpolation CRS to the target geographic CRS. There is then a 3-level loop to create the final set of operations chaining together:

the horizontal transformation from the source geographic CRS to the interpolation CRS

the vertical transformation from the source vertical CRS to the interpolation CRS

the horizontal transformation from the interpolation CRS to the target geographic CRS.

This is implemented by the `createOperationsCompoundToGeog`

method

Example:

```
$ projinfo -s "NAD83(NSRS2007) + NAVD88 height" -t "WGS 84 (G1762)" --spatial-test intersects --summary
Candidate operations found: 21
unknown id, Inverse of NAD83(NSRS2007) to NAVD88 height (1) + NAD83(NSRS2007) to WGS 84 (1) + WGS 84 to WGS 84 (G1762), 3.05 m, USA - CONUS - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (7) + NAD83(HARN) to WGS 84 (1) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS south of 41°N, 95°W to 78°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (7) + NAD83(HARN) to WGS 84 (3) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS south of 41°N, 95°W to 78°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (6) + NAD83(HARN) to WGS 84 (1) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS south of 41°N, 112°W to 95°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (6) + NAD83(HARN) to WGS 84 (3) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS south of 41°N, 112°W to 95°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (2) + NAD83(HARN) to WGS 84 (1) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS north of 41°N, 112°W to 95°W
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (2) + NAD83(HARN) to WGS 84 (3) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS north of 41°N, 112°W to 95°W
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (3) + NAD83(HARN) to WGS 84 (1) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS north of 41°N, 95°W to 78°W
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (3) + NAD83(HARN) to WGS 84 (3) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS north of 41°N, 95°W to 78°W
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (5) + NAD83(HARN) to WGS 84 (1) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS south of 41°N, west of 112°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (5) + NAD83(HARN) to WGS 84 (3) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS south of 41°N, west of 112°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (1) + NAD83(HARN) to WGS 84 (1) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS north of 41°N, west of 112°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (1) + NAD83(HARN) to WGS 84 (3) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS north of 41°N, west of 112°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (4) + NAD83(HARN) to WGS 84 (1) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS north of 41°N, east of 78°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (4) + NAD83(HARN) to WGS 84 (3) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS north of 41°N, east of 78°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (8) + NAD83(HARN) to WGS 84 (1) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS south of 41°N, east of 78°W - onshore
unknown id, Inverse of NAD83(HARN) to NAD83(NSRS2007) (1) + Inverse of NAD83(HARN) to NAVD88 height (8) + NAD83(HARN) to WGS 84 (3) + WGS 84 to WGS 84 (G1762), 3.15 m, USA - CONUS south of 41°N, east of 78°W - onshore
unknown id, Ballpark geographic offset from NAD83(NSRS2007) to NAD83(FBN) + Inverse of NAD83(FBN) to NAVD88 height (1) + Ballpark geographic offset from NAD83(FBN) to WGS 84 (G1762), unknown accuracy, USA - CONUS - onshore, has ballpark transformation
unknown id, Ballpark geographic offset from NAD83(NSRS2007) to NAD83(2011) + Inverse of NAD83(2011) to NAVD88 height (3) + Ballpark geographic offset from NAD83(2011) to WGS 84 (G1762), unknown accuracy, USA - CONUS - onshore, has ballpark transformation
unknown id, Ballpark geographic offset from NAD83(NSRS2007) to NAD83(2011) + Inverse of NAD83(2011) to NAVD88 height (3) + Conversion from NAD83(2011) (geog2D) to NAD83(2011) (geocentric) + Inverse of ITRF2008 to NAD83(2011) (1) + Inverse of WGS 84 (G1762) to ITRF2008 (1) + Conversion from WGS 84 (G1762) (geocentric) to WGS 84 (G1762) (geog2D), unknown accuracy, USA - CONUS - onshore, has ballpark transformation
unknown id, NAD83(NSRS2007) to WGS 84 (1) + WGS 84 to WGS 84 (G1762) + Transformation from NAVD88 height to WGS 84 (G1762) (ballpark vertical transformation, without ellipsoid height to vertical height correction), unknown accuracy, USA - CONUS and Alaska; PRVI, has ballpark transformation
```

## CompoundCRS to CompoundCRS

There is some similarity with the previous paragraph. We first research the vertical transformations between the two vertical CRS.

If there is such a transformation, be it direct, or if both vertical CRS relate to a common intermediate CRS. If it has a registered interpolation geographic CRS, then it is used. Otherwise we fallback to the geographic CRS of the source CRS.

Finally, a 3-level loop to create the final set of operations chaining together:

the horizontal transformation from the source CRS to the interpolation CRS

the vertical transformation

the horizontal transformation from the interpolation CRS to the target CRS.

Example:

$ projinfo -s "NAD27 + NGVD29 height (ftUS)" -t "NAD83 + NAVD88 height" --spatial-test intersects --summary Candidate operations found: 20 unknown id, NGVD29 height (ftUS) to NAVD88 height (3) + NAD27 to NAD83 (1), 0.17 m, USA - CONUS east of 89°W - onshore unknown id, NGVD29 height (ftUS) to NAVD88 height (2) + NAD27 to NAD83 (1), 0.17 m, USA - CONUS 89°W-107°W - onshore unknown id, NGVD29 height (ftUS) to NAVD88 height (1) + NAD27 to NAD83 (1), 0.17 m, USA - CONUS west of 107°W - onshore unknown id, NGVD29 height (ftUS) to NAVD88 height (3) + NAD27 to NAD83 (3), 1.02 m, unknown domain of validity unknown id, NGVD29 height (ftUS) to NAVD88 height (2) + NAD27 to NAD83 (3), 1.02 m, unknown domain of validity unknown id, NGVD29 height (ftUS) to NAVD88 height (1) + NAD27 to NAD83 (3), 1.02 m, unknown domain of validity unknown id, NGVD29 height (ftUS) to NAVD88 height (3) + NAD27 to NAD83 (5), 1.02 m, unknown domain of validity, at least one grid missing unknown id, NGVD29 height (ftUS) to NAVD88 height (3) + NAD27 to NAD83 (6), 1.52 m, unknown domain of validity, at least one grid missing unknown id, NGVD29 height (ftUS) to NAVD88 height (2) + NAD27 to NAD83 (9), 1.52 m, unknown domain of validity, at least one grid missing unknown id, NGVD29 height (ftUS) to NAVD88 height (1) + NAD27 to NAD83 (9), 1.52 m, unknown domain of validity, at least one grid missing unknown id, NGVD29 height (ftUS) to NAVD88 height (3) + Ballpark geographic offset from NAD27 to NAD83, unknown accuracy, USA - CONUS east of 89°W - onshore, has ballpark transformation unknown id, NGVD29 height (ftUS) to NAVD88 height (2) + Ballpark geographic offset from NAD27 to NAD83, unknown accuracy, USA - CONUS 89°W-107°W - onshore, has ballpark transformation unknown id, NGVD29 height (ftUS) to NAVD88 height (1) + Ballpark geographic offset from NAD27 to NAD83, unknown accuracy, USA - CONUS west of 107°W - onshore, has ballpark transformation unknown id, Transformation from NGVD29 height (ftUS) to NAVD88 height (ballpark vertical transformation) + NAD27 to NAD83 (1), unknown accuracy, USA - CONUS including EEZ, has ballpark transformation unknown id, Transformation from NGVD29 height (ftUS) to NAVD88 height (ballpark vertical transformation) + NAD27 to NAD83 (3), unknown accuracy, Canada, has ballpark transformation unknown id, Transformation from NGVD29 height (ftUS) to NAVD88 height (ballpark vertical transformation) + NAD27 to NAD83 (4), unknown accuracy, Canada - NAD27, has ballpark transformation unknown id, Transformation from NGVD29 height (ftUS) to NAVD88 height (ballpark vertical transformation) + NAD27 to NAD83 (5), unknown accuracy, Canada - Quebec, has ballpark transformation, at least one grid missing unknown id, Transformation from NGVD29 height (ftUS) to NAVD88 height (ballpark vertical transformation) + NAD27 to NAD83 (6), unknown accuracy, Canada - Quebec, has ballpark transformation, at least one grid missing unknown id, Transformation from NGVD29 height (ftUS) to NAVD88 height (ballpark vertical transformation) + NAD27 to NAD83 (9), unknown accuracy, Canada - Saskatchewan, has ballpark transformation, at least one grid missing unknown id, Transformation from NGVD29 height (ftUS) to NAVD88 height (ballpark vertical transformation) + Ballpark geographic offset from NAD27 to NAD83, unknown accuracy, World, has ballpark transformation

Otherwise, when there is no such transformation, we decompose into 3 steps:

transform from the source CRS to the geographic 3D CRS corresponding to it

transform from the geographic 3D CRS corresponding to the source CRS to the geographic 3D CRS corresponding to the target CRS

transform from the geographic 3D CRS corresponding to the target CRS to the target CRS.

Example:

$ projinfo -s "WGS 84 + EGM96 height" -t "ETRS89 + Belfast height" --spatial-test intersects --summary Candidate operations found: 7 unknown id, Inverse of WGS 84 to EGM96 height (1) + Inverse of ETRS89 to WGS 84 (1) + ETRS89 to Belfast height (2), 2.014 m, UK - Northern Ireland - onshore unknown id, Inverse of WGS 84 to EGM96 height (1) + Inverse of ETRS89 to WGS 84 (1) + ETRS89 to Belfast height (1), 2.03 m, UK - Northern Ireland - onshore, at least one grid missing unknown id, Inverse of WGS 84 to EGM96 height (1) + Null geographic offset from WGS 84 (geog3D) to WGS 84 (geog2D) + Inverse of OSGB 1936 to WGS 84 (4) + OSGB 1936 to ETRS89 (2) + Null geographic offset from ETRS89 (geog2D) to ETRS89 (geog3D) + ETRS89 to Belfast height (2), 19.044 m, unknown domain of validity unknown id, Inverse of WGS 84 to EGM96 height (1) + Null geographic offset from WGS 84 (geog3D) to WGS 84 (geog2D) + Inverse of OSGB 1936 to WGS 84 (2) + OSGB 1936 to ETRS89 (2) + Null geographic offset from ETRS89 (geog2D) to ETRS89 (geog3D) + ETRS89 to Belfast height (2), 11.044 m, unknown domain of validity unknown id, Inverse of WGS 84 to EGM96 height (1) + Null geographic offset from WGS 84 (geog3D) to WGS 84 (geog2D) + Inverse of TM75 to WGS 84 (2) + TM75 to ETRS89 (3) + Null geographic offset from ETRS89 (geog2D) to ETRS89 (geog3D) + ETRS89 to Belfast height (2), 2.424 m, UK - Northern Ireland - onshore, at least one grid missing unknown id, Inverse of WGS 84 to EGM96 height (1) + Null geographic offset from WGS 84 (geog3D) to WGS 84 (geog2D) + Inverse of TM75 to WGS 84 (2) + TM75 to ETRS89 (3) + Null geographic offset from ETRS89 (geog2D) to ETRS89 (geog3D) + ETRS89 to Belfast height (1), 2.44 m, UK - Northern Ireland - onshore, at least one grid missing unknown id, Inverse of WGS 84 to EGM96 height (1) + Null geographic offset from WGS 84 (geog3D) to WGS 84 (geog2D) + Inverse of OSGB 1936 to WGS 84 (4) + OSGB 1936 to ETRS89 (2) + Null geographic offset from ETRS89 (geog2D) to ETRS89 (geog3D) + ETRS89 to Belfast height (1), 19.06 m, unknown domain of validity, at least one grid missing

This is implemented by the `createOperationsCompoundToCompound`

method

## When the source or target CRS is a BoundCRS

The BoundCRS concept is an hybrid concept where a CRS is linked to a transformation
from it to a hub CRS, typically WGS 84. This is a long-time practice in PROJ.4
strings with the `+towgs84`

, `+nadgrids`

and `+geoidgrids`

keywords, or the
`TOWGS84[]`

node of WKT 1. When encountering those attributes when parsing
a CRS string, PROJ will create a BoundCRS object capturing this transformation.
A BoundCRS object can also be provided with a WKT2 string, and in that case with
a hub CRS being potentially different from WGS 84.

Let's consider the case of a transformation between a BoundCRS ("+proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 +y_0=-100000 +ellps=airy +towgs84=446.448,-125.157,542.06,0.15,0.247,0.842,-20.489 +units=m" which used to be the PROJ.4 definition of "OSGB 1936 / British National Grid") and a target Geographic CRS, ETRS89.

We apply the following steps:

transform from the base of the source CRS (that is the CRS wrapped by BoundCRS, here a ProjectedCRS) to the geographic CRS of this base CRS

apply the transformation of the BoundCRS to go from the geographic CRS of this base CRS to the hub CRS of the BoundCRS, in that instance WGS 84.

apply a transformation from the hub CRS to the target CRS.

This is implemented by the `createOperationsBoundToGeog`

method

Example:

```
$ projinfo -s "+proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 +y_0=-100000 +ellps=airy +towgs84=446.448,-125.157,542.06,0.15,0.247,0.842,-20.489 +units=m +type=crs" -t ETRS89 -o PROJ
Candidate operations found: 1
-------------------------------------
Operation No. 1:
unknown id, Inverse of unknown + Transformation from unknown to WGS84 + Inverse of ETRS89 to WGS 84 (1), unknown accuracy, Europe - ETRS89
PROJ string:
+proj=pipeline +step +inv +proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 +y_0=-100000 +ellps=airy +step +proj=push +v_3 +step +proj=cart +ellps=airy +step +proj=helmert +x=446.448 +y=-125.157 +z=542.06 +rx=0.15 +ry=0.247 +rz=0.842 +s=-20.489 +convention=position_vector +step +inv +proj=cart +ellps=GRS80 +step +proj=pop +v_3 +step +proj=unitconvert +xy_in=rad +xy_out=deg +step +proj=axisswap +order=2,1
```

There are other situations with BoundCRS, involving vertical transformations, or transforming to other objects than a geographic CRS, but the curious reader will have to inspect the code for the actual gory details.

## Using Derived Projected for source or target CRS

The WKT2 tag `DERIVEDPROJCRS`

can be useful to define a customized CRS
adding a `DERIVINGCONVERSION`

to apply a conversion on top of the projection.
It can be also done inside a Compound CRS.
One use case is to describe a local CRS produced in a site calibration,
as explained in [JimenezShaw2023].

Using the WKT2 from that paper (stored in wkt2.txt file for readability) we would get this:

```
projinfo -s EPSG:6319 -t "`cat wkt2.txt`" -o proj
Candidate operations found: 1
-------------------------------------
Operation No. 1:
unknown id, Inverse of Transformation from Ellipsoid (metre) to NAD83(2011) (ballpark vertical transformation, without ellipsoid height to vertical height correction) + Conv Vertical Offset and Slope + Transverse Mercator + Affine transformation as PROJ-based, unknown accuracy, World, has ballpark transformation
PROJ string:
+proj=pipeline
+step +proj=axisswap +order=2,1
+step +proj=unitconvert +xy_in=deg +xy_out=rad
+step +proj=vertoffset +lat_0=41.2305352787143 +lon_0=-73.1815861874286
+dh=31.0121985701957 +slope_lat=-6.12572852418232
+slope_lon=-2.67487863214139
+step +proj=tmerc +lat_0=41.2305352787143 +lon_0=-73.1815861874286 +k=1 +x_0=0
+y_0=0 +ellps=GRS80
+step +proj=affine +xoff=265262.95287 +yoff=196619.27389 +s11=1.00003994119
+s12=0.00548156923529 +s21=-0.00548156923529 +s22=1.00003994119
```