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<section id="triangulated-irregular-network-based-transformation">
<span id="tinshift"></span><h1>Triangulated Irregular Network based transformation<a class="headerlink" href="#triangulated-irregular-network-based-transformation" title="Permalink to this headline">¶</a></h1>
<div class="versionadded">
<p><span class="versionmodified added">New in version 7.2.0.</span></p>
</div>
<table class="docutils align-default">
<colgroup>
<col style="width: 24%" />
<col style="width: 76%" />
</colgroup>
<tbody>
<tr class="row-odd"><td><p><strong>Alias</strong></p></td>
<td><p>tinshift</p></td>
</tr>
<tr class="row-even"><td><p><strong>Input type</strong></p></td>
<td><p>Projected or geographic coordinates (horizontal), meters (vertical)</p></td>
</tr>
<tr class="row-odd"><td><p><strong>Output type</strong></p></td>
<td><p>Projected or geographic coordinates (horizontal), meters (vertical)</p></td>
</tr>
<tr class="row-even"><td><p><strong>Domain</strong></p></td>
<td><p>2D or 3D</p></td>
</tr>
<tr class="row-odd"><td><p><strong>Available forms</strong></p></td>
<td><p>Forward and inverse</p></td>
</tr>
</tbody>
</table>
<p>The <code class="docutils literal notranslate"><span class="pre">tinshift</span></code> transformation takes one mandatory
argument, <code class="docutils literal notranslate"><span class="pre">file</span></code>, that points to a JSON file, which contains the
triangulation and associated metadata. Input and output coordinates must be
geographic or projected coordinates.
Depending on the content of the JSON file, horizontal, vertical or both
components of the coordinates may be transformed.</p>
<p>The transformation is invertible, with the same computational complexity than
the forward transformation.</p>
<section id="parameters">
<h2>Parameters<a class="headerlink" href="#parameters" title="Permalink to this headline">¶</a></h2>
<section id="required">
<h3>Required<a class="headerlink" href="#required" title="Permalink to this headline">¶</a></h3>
<dl class="std option">
<dt class="sig sig-object std" id="cmdoption-arg-file">
<span class="sig-name descname"><span class="pre">+file</span></span><span class="sig-prename descclassname"><span class="pre">=<filename></span></span><a class="headerlink" href="#cmdoption-arg-file" title="Permalink to this definition">¶</a></dt>
<dd><p>Filename to the JSON file for the TIN.</p>
</dd></dl>
</section>
</section>
<section id="example">
<h2>Example<a class="headerlink" href="#example" title="Permalink to this headline">¶</a></h2>
<p>Transforming a point with the Finland EPSG:2393 (“KKJ / Finland Uniform
Coordinate System”) projected CRS to EPSG:3067 (“ETRS89 / TM35FIN(E,N)”)</p>
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>$ echo 3210000.0000 6700000.0000 0 2020 | cct +proj=tinshift +file=./triangulation_kkj.json
209948.3217 6697187.0009 0.0000 2020
</pre></div>
</div>
<section id="algorithm">
<h3>Algorithm<a class="headerlink" href="#algorithm" title="Permalink to this headline">¶</a></h3>
<p>Internally, <code class="docutils literal notranslate"><span class="pre">tinshift</span></code> ingest the whole file into memory. It is considered that
triangulation should be small enough for that.</p>
<p>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.</p>
<p>To determine if a point falls into a triangle, one computes its 3
<a class="reference external" href="https://en.wikipedia.org/wiki/Barycentric_coordinate_system#Conversion_between_barycentric_and_Cartesian_coordinates">barycentric coordinates</a>
from its projected coordinates, <span class="math notranslate nohighlight">\(\lambda_i\)</span> for <span class="math notranslate nohighlight">\(i=1,2,3\)</span>.
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.</p>
<p>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 (<span class="math notranslate nohighlight">\(Xt_i\)</span> and <span class="math notranslate nohighlight">\(Yt_i\)</span>):</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}X_{target} = Xt_1 * \lambda_1 + Xt_2 * \lambda_2 + Xt_3 * \lambda_3\\Y_{target} = Yt_1 * \lambda_1 + Yt_2 * \lambda_2 + Yt_3 * \lambda_3\end{aligned}\end{align} \]</div>
<p>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</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}X_{target} = A + X_{source} * B + Y_{source} * C\\Y_{target} = D + X_{source} * E + Y_{source} * F\end{aligned}\end{align} \]</div>
<p>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:</p>
<div class="math notranslate nohighlight">
\[ \begin{align}\begin{aligned}Xt_i = A + Xs_i * B + Ys_i * C\\Yt_i = D + Xs_i * E + Ys_i * F\end{aligned}\end{align} \]</div>
<p>Similarly for a vertical coordinate transformation, where <span class="math notranslate nohighlight">\(Zoff_i\)</span> is the vertical
offset at each vertex of the triangle:</p>
<div class="math notranslate nohighlight">
\[Z_{target} = Z_{source} + Zoff_1 * \lambda_1 + Zoff_2 * \lambda_2 + Zoff_3 * \lambda_3\]</div>
</section>
<section id="constraints-on-the-triangulation">
<h3>Constraints on the triangulation<a class="headerlink" href="#constraints-on-the-triangulation" title="Permalink to this headline">¶</a></h3>
<p>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)</p>
</section>
<section id="file-format">
<h3>File format<a class="headerlink" href="#file-format" title="Permalink to this headline">¶</a></h3>
<p>The triangulation is stored in a text-based format, using JSON as a serialization.</p>
<p>Below a minimal example, from the KKJ to ETRS89 transformation, with just a
single triangle:</p>
<div class="highlight-json notranslate"><div class="highlight"><pre><span></span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="nt">"file_type"</span><span class="p">:</span><span class="w"> </span><span class="s2">"triangulation_file"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"format_version"</span><span class="p">:</span><span class="w"> </span><span class="s2">"1.0"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"name"</span><span class="p">:</span><span class="w"> </span><span class="s2">"Name"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"version"</span><span class="p">:</span><span class="w"> </span><span class="s2">"Version"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"publication_date"</span><span class="p">:</span><span class="w"> </span><span class="s2">"2018-07-01T00:00:00Z"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"license"</span><span class="p">:</span><span class="w"> </span><span class="s2">"Creative Commons Attribution 4.0 International"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"description"</span><span class="p">:</span><span class="w"> </span><span class="s2">"Test triangulation"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"authority"</span><span class="p">:</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="nt">"name"</span><span class="p">:</span><span class="w"> </span><span class="s2">"Authority name"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"url"</span><span class="p">:</span><span class="w"> </span><span class="s2">"http://example.com"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"address"</span><span class="p">:</span><span class="w"> </span><span class="s2">"Adress"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"email"</span><span class="p">:</span><span class="w"> </span><span class="s2">"test@example.com"</span><span class="w"></span>
<span class="w"> </span><span class="p">},</span><span class="w"></span>
<span class="w"> </span><span class="nt">"links"</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="w"></span>
<span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="nt">"href"</span><span class="p">:</span><span class="w"> </span><span class="s2">"https://example.com/about.html"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"rel"</span><span class="p">:</span><span class="w"> </span><span class="s2">"about"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"type"</span><span class="p">:</span><span class="w"> </span><span class="s2">"text/html"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"title"</span><span class="p">:</span><span class="w"> </span><span class="s2">"About"</span><span class="w"></span>
<span class="w"> </span><span class="p">},</span><span class="w"></span>
<span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="nt">"href"</span><span class="p">:</span><span class="w"> </span><span class="s2">"https://example.com/download"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"rel"</span><span class="p">:</span><span class="w"> </span><span class="s2">"source"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"type"</span><span class="p">:</span><span class="w"> </span><span class="s2">"application/zip"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"title"</span><span class="p">:</span><span class="w"> </span><span class="s2">"Authoritative source"</span><span class="w"></span>
<span class="w"> </span><span class="p">},</span><span class="w"></span>
<span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="nt">"href"</span><span class="p">:</span><span class="w"> </span><span class="s2">"https://creativecommons.org/licenses/by/4.0/"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"rel"</span><span class="p">:</span><span class="w"> </span><span class="s2">"license"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"type"</span><span class="p">:</span><span class="w"> </span><span class="s2">"text/html"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"title"</span><span class="p">:</span><span class="w"> </span><span class="s2">"Creative Commons Attribution 4.0 International license"</span><span class="w"></span>
<span class="w"> </span><span class="p">},</span><span class="w"></span>
<span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="nt">"href"</span><span class="p">:</span><span class="w"> </span><span class="s2">"https://example.com/metadata.xml"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"rel"</span><span class="p">:</span><span class="w"> </span><span class="s2">"metadata"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"type"</span><span class="p">:</span><span class="w"> </span><span class="s2">"application/xml"</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="nt">"title"</span><span class="p">:</span><span class="w"> </span><span class="s2">" ISO 19115 XML encoded metadata regarding the deformation model"</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="w"> </span><span class="p">],</span><span class="w"></span>
<span class="w"> </span><span class="nt">"transformed_components"</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="w"> </span><span class="s2">"horizontal"</span><span class="w"> </span><span class="p">],</span><span class="w"></span>
<span class="w"> </span><span class="nt">"vertices_columns"</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="w"> </span><span class="s2">"source_x"</span><span class="p">,</span><span class="w"> </span><span class="s2">"source_y"</span><span class="p">,</span><span class="w"> </span><span class="s2">"target_x"</span><span class="p">,</span><span class="w"> </span><span class="s2">"target_y"</span><span class="w"> </span><span class="p">],</span><span class="w"></span>
<span class="w"> </span><span class="nt">"triangles_columns"</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="w"> </span><span class="s2">"idx_vertex1"</span><span class="p">,</span><span class="w"> </span><span class="s2">"idx_vertex2"</span><span class="p">,</span><span class="w"> </span><span class="s2">"idx_vertex3"</span><span class="w"> </span><span class="p">],</span><span class="w"></span>
<span class="w"> </span><span class="nt">"vertices"</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="w"> </span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">49</span><span class="p">,</span><span class="mf">2.1</span><span class="p">,</span><span class="mf">49.1</span><span class="p">],</span><span class="w"> </span><span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">50</span><span class="p">,</span><span class="mf">3.1</span><span class="p">,</span><span class="mf">50.1</span><span class="p">],</span><span class="w"> </span><span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="w"> </span><span class="mi">50</span><span class="p">,</span><span class="w"> </span><span class="mf">2.1</span><span class="p">,</span><span class="mf">50.1</span><span class="p">]</span><span class="w"> </span><span class="p">],</span><span class="w"></span>
<span class="w"> </span><span class="nt">"triangles"</span><span class="p">:</span><span class="w"> </span><span class="p">[</span><span class="w"> </span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="p">]</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
</pre></div>
</div>
<p>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 <code class="docutils literal notranslate"><span class="pre">vertices</span></code> and <code class="docutils literal notranslate"><span class="pre">triangles</span></code> 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 <code class="docutils literal notranslate"><span class="pre">vertices</span></code> array.</p>
<p>More formally, the specific items for the triangulation file are:</p>
<dl class="simple">
<dt>input_crs</dt><dd><p>String identifying the CRS of source coordinates
in the vertices. Typically <code class="docutils literal notranslate"><span class="pre">EPSG:XXXX</span></code>. 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
(<code class="docutils literal notranslate"><span class="pre">KKJ</span> <span class="pre">/</span> <span class="pre">Finland</span> <span class="pre">Uniform</span> <span class="pre">Coordinate</span> <span class="pre">System</span></code>). 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.</p>
</dd>
<dt>output_crs</dt><dd><p>String identifying the CRS of target coordinates in the vertices.
Typically <code class="docutils literal notranslate"><span class="pre">EPSG:XXXX</span></code>. 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.</p>
</dd>
<dt>transformed_components</dt><dd><p>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.</p>
</dd>
<dt>fallback_strategy</dt><dd><p>String identifying how to treat points that do not fall into any of the
specified triangles. This item is available for <code class="docutils literal notranslate"><span class="pre">format_version</span></code> >= 1.1.
Possible values are <code class="docutils literal notranslate"><span class="pre">none</span></code>, <code class="docutils literal notranslate"><span class="pre">nearest_side</span></code> and <code class="docutils literal notranslate"><span class="pre">nearest_centroid</span></code>. The
default is <code class="docutils literal notranslate"><span class="pre">none</span></code> and signifies, that points that fall outside the
specified triangles are not transformed. This is also the behaviour for
<code class="docutils literal notranslate"><span class="pre">format_version</span></code> before 1.1. If <code class="docutils literal notranslate"><span class="pre">fallback_strategy</span></code> is set to
<code class="docutils literal notranslate"><span class="pre">nearest_side</span></code>, then points that do not fall into any triangle are
transformed according to the triangle closest to them by euclidean distance.
If <code class="docutils literal notranslate"><span class="pre">fallback_strategy</span></code> is set to <code class="docutils literal notranslate"><span class="pre">nearest_centroid</span></code>, then points that do
not fall into any triangle are transformed according to the triangle with the
closest centroid (intersection of its medians).</p>
</dd>
<dt>vertices_columns</dt><dd><p>Specify the name of the columns of the rows in the <code class="docutils literal notranslate"><span class="pre">vertices</span></code>
array. There must be exactly as many elements in <code class="docutils literal notranslate"><span class="pre">vertices_columns</span></code> as in a
row of <code class="docutils literal notranslate"><span class="pre">vertices</span></code>. The following names have a special meaning: <code class="docutils literal notranslate"><span class="pre">source_x</span></code>,
<code class="docutils literal notranslate"><span class="pre">source_y</span></code>, <code class="docutils literal notranslate"><span class="pre">target_x</span></code>, <code class="docutils literal notranslate"><span class="pre">target_y</span></code>, <code class="docutils literal notranslate"><span class="pre">source_z</span></code>, <code class="docutils literal notranslate"><span class="pre">target_z</span></code> and
<code class="docutils literal notranslate"><span class="pre">offset_z</span></code>. <code class="docutils literal notranslate"><span class="pre">source_x</span></code> and <code class="docutils literal notranslate"><span class="pre">source_y</span></code> are compulsory.
<code class="docutils literal notranslate"><span class="pre">source_x</span></code> is for the source longitude (in degree) or easting.
<code class="docutils literal notranslate"><span class="pre">source_y</span></code> is for the source latitude (in degree) or northing.
<code class="docutils literal notranslate"><span class="pre">target_x</span></code> and <code class="docutils literal notranslate"><span class="pre">target_y</span></code> are compulsory when <code class="docutils literal notranslate"><span class="pre">horizontal</span></code> is specified
in <code class="docutils literal notranslate"><span class="pre">transformed_components</span></code>. (<code class="docutils literal notranslate"><span class="pre">source_z</span></code> and <code class="docutils literal notranslate"><span class="pre">target_z</span></code>) or
<code class="docutils literal notranslate"><span class="pre">offset_z</span></code> are compulsory when <code class="docutils literal notranslate"><span class="pre">vertical</span></code> is specified in <code class="docutils literal notranslate"><span class="pre">transformed_components</span></code></p>
</dd>
<dt>triangles_columns</dt><dd><p>Specify the name of the columns of the rows in the
<code class="docutils literal notranslate"><span class="pre">triangles</span></code> array. There must be exactly as many elements in <code class="docutils literal notranslate"><span class="pre">triangles_columns</span></code>
as in a row of <code class="docutils literal notranslate"><span class="pre">triangles</span></code>. The following names have a special meaning:
<code class="docutils literal notranslate"><span class="pre">idx_vertex1</span></code>, <code class="docutils literal notranslate"><span class="pre">idx_vertex2</span></code>, <code class="docutils literal notranslate"><span class="pre">idx_vertex3</span></code>. They are compulsory.</p>
</dd>
<dt>vertices</dt><dd><p>An array whose items are themselves arrays with as many columns as
described in <code class="docutils literal notranslate"><span class="pre">vertices_columns</span></code>.</p>
</dd>
<dt>triangles</dt><dd><p>An array whose items are themselves arrays with as many columns as
described in <code class="docutils literal notranslate"><span class="pre">triangles_columns</span></code>.
The value of the <code class="docutils literal notranslate"><span class="pre">idx_vertexN</span></code> columns must be indices
(between 0 and len(<code class="docutils literal notranslate"><span class="pre">vertices</span></code>-1) of items of the <code class="docutils literal notranslate"><span class="pre">vertices</span></code> array.</p>
</dd>
</dl>
<p>A <a class="reference external" href="https://proj.org/schemas/triangulation.schema.json">JSON schema</a> is available
for this file format.</p>
</section>
</section>
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