Triangulated Irregular Network based transformation

Added in version 7.2.0.

Alias	tinshift
Input type	Projected or geographic coordinates (horizontal), meters (vertical)
Output type	Projected or geographic coordinates (horizontal), meters (vertical)
Domain	2D or 3D
Available forms	Forward and inverse

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

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

Parameters

Required

+file=<filename>

Filename or URL to the JSON or GeoPackage file for the TIN.

Example

Transforming a point with the Finland EPSG:2393 ("KKJ / Finland Uniform Coordinate System") projected CRS to EPSG:3067 ("ETRS89 / TM35FIN(E,N)")

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

209948.3217     6697187.0009    0.0000     2020

Algorithm

Internally, tinshift ingest the whole file into memory. It is considered that triangulation should be small enough for that.

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.

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:

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

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)

JSON TIN File format

See Triangulated irregular network (TIN) JSON format.

GeoPackage TIN File format

Added in version 9.9.0.

See Triangulated irregular network (TIN) GeoPackage format.