# PROJ RFC 6: Triangulation-based transformations¶

Author

Even Rouault

Contact

even.rouault@spatialys.com

Status

Implementation target

PROJ 7.2

Last Updated

2020-09-02

## Summary¶

This RFC adds a new transformation method, tinshift (TIN stands for Triangulated Irregular Network)

The motivation for this work is to be able to handle the official transformations created by National Land Survey of Finland, for:

• horizontal transformation between the KKJ and ETRS89 horizontal datums

• vertical transformations between N43 and N60 heights, and N60 and N2000 heights.

Such transformations are somehow related to traditional grid-based transformations, except that the correction values are hold by the vertices of the triangulation, instead of being at nodes of a grid.

Triangulation are in a number of cases the initial product of a geodesic adjustment, with grids being a derived product. The Swiss grids have for example derived products of an original triangulation.

Grid-based transformations remain very convenient to use because accessing correction values is really easy and efficient, so triangulation-based transformations are not meant as replacing them, but more about it being a complement, that is sometimes necessary to be able to replicate the results of a officially vetted transformation to a millimetric or better precision (speaking here about reproducibility of numeric results, rather than the physical accuracy of the transformation that might rather be centimetric). It is always possible to approach the result of the triangulation with a grid, but that may require to adopt a small grid step, and thus generate a grid that can be much larger than the original triangulation.

## Details¶

### Transformation¶

A new transformation method, tinshift, is added. It takes one mandatory argument, file, that points to a JSON file, which contains the triangulation and associated metadata. Input and output coordinates must be geographic or projected. Depending on the content of the JSON file, horizontal, vertical or both components of the coordinates may be transformed.

The transformation is used like:

\$ echo 3210000.0000 6700000.0000 0 2020 | cct +proj=tinshift +file=./triangulation_kkj.json

209948.3217     6697187.0009    0.0000     2020


The transformation is invertible, with the same computational complexity than the forward transformation.

### Algorithm¶

Internally, tinshift ingest the whole file into memory. It is considered that triangulation should be small enough for that. The above mentioned KKJ to ETRS89 triangulation fits into 65 KB of JSON, for 1449 triangles and 767 vertices.

When a point is transformed, one must find the triangle into which it falls into. Instead of iterating over all triangles, we build a in-memory quadtree to speed-up the identification of candidates triangles. On the above mentioned KKJ -> ETRS89 triangulation, this speeds up the whole transformation by a factor of 10. The quadtree structure is a very good compromise between the performance gain it brings and the simplicity of its implementation (we have ported the implementation coming from GDAL, inherit from the one used for shapefile .spx spatial indices).

To determine if a point falls into a triangle, one computes its 3 barycentric coordinates from its projected coordinates, $$\lambda_i$$ for $$i=1,2,3$$. They are real values (in the [0,1] range for a point inside the triangle), giving the weight of each of the 3 vertices of the triangles.

Once those weights are known, interpolating the target horizontal coordinate is a matter of doing the linear combination of those weights with the target horizontal coordinates at the 3 vertices of the triangle ($$Xt_i$$ and $$Yt_i$$):

\begin{align}\begin{aligned}X_{target} = Xt_1 * \lambda_1 + Xt_2 * \lambda_2 + Xt_3 * \lambda_3\\Y_{target} = Yt_1 * \lambda_1 + Yt_2 * \lambda_2 + Yt_3 * \lambda_3\end{aligned}\end{align}

This interpolation is exact at the vertices of the triangulation, and has linear properties inside each triangle. It is completely equivalent to other formulations of triangular interpolation, such as

\begin{align}\begin{aligned}X_{target} = A + X_{source} * B + Y_{source} * C\\Y_{target} = D + X_{source} * E + Y_{source} * F\end{aligned}\end{align}

where the A, B, C, D, E, F constants (for a given triangle) are found by solving the 2 systems of 3 linear equations, constraint by the source and target coordinate pairs of the 3 vertices of the triangle:

\begin{align}\begin{aligned}Xt_i = A + Xs_i * B + Ys_i * C\\Yt_i = D + Xs_i * E + Ys_i * F\end{aligned}\end{align}

Note

From experiments, the interpolation using barycentric coordinates is slightly more numerically robust when interpolating projected coordinates of amplitude of the order of 1e5 / 1e6, due to computations involving differences of coordinates. Whereas the formulation with the A, B, C, D, E, F tends to have big values for the A and D constants, and values clause to 0 for C and E, and close to 1 for B and F. However, the difference between the two approaches is negligible for practical purposes (below micrometre precision)

Similarly for a vertical coordinate transformation, where $$Zoff_i$$ is the vertical offset at each vertex of the triangle:

$Z_{target} = Z_{source} + Zoff_1 * \lambda_1 + Zoff_2 * \lambda_2 + Zoff_3 * \lambda_3$

### Constraints on the triangulation¶

No check is done on the consistence of the triangulation. It is highly recommended that triangles do not overlap each other (when considering the source coordinates or the forward transformation, or the target coordinates for the inverse transformation), otherwise which triangle will be selected is unspecified. Besides that, the triangulation does not need to have particular properties (like being a Delaunay triangulation)

### File format¶

To the best of our knowledge, there are no established file formats to convey geodetic transformations as triangulations. Potential similar formats to store TINs are ITF or XMS. Both of them would need to be extended in order to handle datum shift information, since they are both intended for mostly DEM use.

We thus propose a text-based format, using JSON as a serialization. Using a text-based format could potentially be thought as a limitation performance-wise compared to binary formats, but for the size of triangulations considered (a few thousands triangles / vertices), there is no issue. Loading such file is a matter of 20 milliseconds or so. For reference, loading a triangulation of about 115 000 triangles and 71 000 vertices takes 450 ms.

Using JSON provides generic formatting and parsing rules, and convenience to create it from Python script for examples. This could also be easily generated “at hand” by non-JSON aware writers.

For generic metadata, we reuse closely what has been used for the Deformation model master file

Below a minimal example, from the KKJ to ETRS89 transformation, with just a single triangle:

{
"file_type": "triangulation_file",
"format_version": "1.0",
"name": "Name",
"version": "Version",
"publication_date": "2018-07-01T00:00:00Z",
"description": "Test triangulation",
"authority": {
"name": "Authority name",
"url": "http://example.com",
"email": "test@example.com"
},
{
"type": "text/html",
},
{
"rel": "source",
"type": "application/zip",
"title": "Authoritative source"
},
{
"type": "text/html",
},
{
"type": "application/xml",
"title": " ISO 19115 XML encoded metadata regarding the triangulation"
}
],
"input_crs": "EPSG:2393",
"target_crs": "EPSG:3067",
"transformed_components": [ "horizontal" ],
"vertices_columns": [ "source_x", "source_y", "target_x", "target_y" ],
"triangles_columns": [ "idx_vertex1", "idx_vertex2", "idx_vertex3" ],
"vertices": [ [3244102.707, 6693710.937, 244037.137, 6690900.686],
[3205290.722, 6715311.822, 205240.895, 6712492.577],
[3218328.492, 6649538.429, 218273.648, 6646745.973] ],
"triangles": [ [0, 1, 2] ]
}


So after the generic metadata, we define the input and output CRS (informative only), and that the transformation affects horizontal components of coordinates. We name the columns of the vertices and triangles arrays. We defined the source and target coordinates of each vertex, and define a triangle by referring to the index of its vertices in the vertices array.

More formally, the specific items for the triangulation file are:

input_crs

String identifying the CRS of source coordinates in the vertices. Typically EPSG:XXXX. If the transformation is for vertical component, this should be the code for a compound CRS (can be EPSG:XXXX+YYYY where XXXX is the code of the horizontal CRS and YYYY the code of the vertical CRS). For example, for the KKJ->ETRS89 transformation, this is EPSG:2393 (KKJ / Finland Uniform Coordinate System). The input coordinates are assumed to be passed in the “normalized for visualisation” / “GIS friendly” order, that is longitude, latitude for geographic coordinates and easting, northing for projected coordinates.

output_crs

String identifying the CRS of target coordinates in the vertices. Typically EPSG:XXXX. If the transformation is for vertical component, this should be the code for a compound CRS (can be EPSG:XXXX+YYYY where XXXX is the code of the horizontal CRS and YYYY the code of the vertical CRS). For example, for the KKJ->ETRS89 transformation, this is EPSG:3067 ("ETRS89 / TM35FIN(E,N)"). The output coordinates will be returned in the “normalized for visualisation” / “GIS friendly” order, that is longitude, latitude for geographic coordinates and easting, northing for projected coordinates.

transformed_components

Array which may contain one or two strings: “horizontal” when horizontal components of the coordinates are transformed and/or “vertical” when the vertical component is transformed.

vertices_columns

Specify the name of the columns of the rows in the vertices array. There must be exactly as many elements in vertices_columns as in a row of vertices. The following names have a special meaning: source_x, source_y, target_x, target_y, source_z, target_z and offset_z. source_x and source_y are compulsory. source_x is for the source longitude (in degree) or easting. source_y is for the source latitude (in degree) or northing. target_x and target_y are compulsory when horizontal is specified in transformed_components. (source_z and target_z) or offset_z are compulsory when vertical is specified in transformed_components

triangles_columns

Specify the name of the columns of the rows in the triangles array. There must be exactly as many elements in triangles_columns as in a row of triangles. The following names have a special meaning: idx_vertex1, idx_vertex2, idx_vertex3. They are compulsory.

vertices

An array whose items are themselves arrays with as many columns as described in vertices_columns.

triangles

An array whose items are themselves arrays with as many columns as described in triangles_columns. The value of the idx_vertexN columns must be indices (between 0 and len(vertices-1) of items of the vertices array.

### Code impacts¶

The following new files are added in src/transformations:

• tinshift.cpp: PROJ specific code for defining the new operation. Takes care of the input and output coordinate conversions (between input_crs and triangulation_source_crs, and triangulation_target_crs and output_crs), when needed.

• tinshift.hpp: Header-based implementation. This file contains the API.

• tinshift_exceptions.hpp: Exceptions that can be raised during file parsing

This is the approach that has been followed for the deformation model implementation, and which makes it easier to do unit test.

### Performance indications¶

Tested on Intel(R) Core(TM) i7-6700HQ CPU @ 2.60GHz, transforming 4 million points

For the KKJ to ETRS89 transformation (1449 triangles and 767 vertices), 4.4 million points / sec can be transformed.

For comparison, the Helmert-based KKJ to ETRS89 transformation operates at 1.6 million points / sec.

A triangulation with about 115 000 triangles and 71 000 vertices operates at 2.2 million points / sec (throughput on more points would be better since the initial loading of the triangulation is non-negligible here)

## Backward compatibility¶

New functionality fully backward compatible.

## Testing¶

The PROJ test suite will be enhanced to test the new transformation, with a new .gie file, and a C++ unit test to test at a lower level.

## Proposed implementation¶

An initial implementation is available at https://github.com/rouault/PROJ/tree/tinshift

The RFC was adopted on 2020-09-02 with +1’s from the following PSC members

• Kristian Evers

• Charles Karney

• Thomas Knudsen

• Even Rouault

## Funding¶

This work is funded by National Land Survey of Finland