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<p class="MsoNormal">Dear list members,<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><span lang="EN-GB">The implementation of </span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+proj=horner</span><span lang="EN-GB"> is nice. I used it to reproduce a 99 year old polynomial transformation of an even older
Dutch CRS published in 1827. However, I encountered some problems with the documentation on
<a href="https://proj.org/en/9.3/operations/transformations/horner.html">https://proj.org/en/9.3/operations/transformations/horner.html</a>. The order of the coefficients in the formulas and in the description of the PROJ parameters
</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">(+fwd_u</span><span lang="EN-GB"> and
</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+fwd_u</span><span lang="EN-GB">) seem mutually inconsistent.
<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal"><b><span lang="EN-GB">Problem 1:</span></b><span lang="EN-GB"> The matrix formula (after formula 3) seems inconsistent with formula (1) and the actual implementation.<o:p></o:p></span></p>
<p class="MsoListParagraph"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">\begin{bmatrix}<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas"> u_{<span style="color:red">0,1</span>} + u_{<span style="color:red">0,2</span>} U + ... & u_{<span style="color:red">1,0</span>} + u_{1,1}
U + u_{<span style="color:red">2,0</span>} V + ... \\<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas"> v_{<span style="color:red">1,0</span>} + v_{1,1} V + v_{<span style="color:red">2,0</span>} U + ... & v_{<span style="color:red">0,1</span>}
+ v_{<span style="color:red">0,2</span>} V \\<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">\end{bmatrix}<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB">should be changed to:<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">\begin{bmatrix}<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">
</span><span style="font-size:10.0pt;font-family:Consolas">u_{1,0} + u_{2,0} U + ... & u_{0,1} + u_{1,1} U + u_{0,2} V + ... \\<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span style="font-size:10.0pt;font-family:Consolas"> v_{0,1} + v_{1,1} V + v_{0,2} U + ... & v_{1,0} + v_{2,0} V \\<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span style="font-size:10.0pt;font-family:Consolas">\end{bmatrix}<o:p></o:p></span></p>
<p class="MsoNormal"><b><span lang="EN-GB"><o:p> </o:p></span></b></p>
<p class="MsoNormal"><b><span lang="EN-GB">Problem 2:</span></b><span lang="EN-GB"> The description of the parameters seems inconsistent with formula (1) and the actual implementation.<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+fwd_u=<u_<span style="color:red">11</span>,u_<span style="color:red">12</span>,...,u_ij,..,u_<span style="color:red">mn</span>><o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB">and<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+fwd_v=<v_<span style="color:red">11</span>,v_<span style="color:red">12</span>,...,v_ij,..,v_<span style="color:red">mn</span>><o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB">should be changed</span><span lang="EN-GB">
</span>to:<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span style="font-size:10.0pt;font-family:Consolas">+fwd_u=<u_00,u_10,...,u_ij,..,u_nn><o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB">and<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+fwd_v=<v_00,v_01,...,v_ij,..,v_nn><o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB">As a result it was difficult to determine the correct order of the coefficients for
</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+fwd_u</span><span lang="EN-GB"> and
</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+fwd_v</span><span lang="EN-GB"> parameters. I think it would be clarifying to add a table for the order of the coefficients in the documentation (example for
</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+deg=3</span><span lang="EN-GB">):<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB">Order of coefficients of</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas"> +fwd_u:<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas"> | x^0 | x^1 | x^2 | x^3<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^0 | 1 | 2 | 3 | 4
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^1 | 5 | 6 | 7 | -
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^2 | 8 | 9 | - | -
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^3 | 10 | - | - | -
<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB">Order of coefficients of</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas"> +fwd_v:<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas"> | x^0 | x^1 | x^2 | x^3<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^0 | 1 | 5 | 8 | 10
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^1 | 2 | 6 | 9 | -
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^2 | 3 | 7 | - | -
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^3 | 4 | - | - | -
<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal"><b><span lang="EN-GB">Suggestion for a different implementation:</span></b><span lang="EN-GB"> Actually, I would have preferred a different order of the coefficients in the implementation, in a way that the order would be the same for
</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+fwd_u</span><span lang="EN-GB"> and
</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+fwd_v</span><span lang="EN-GB">, and it would keep the first coefficients the same when increasing the degree (</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">+deg</span><span lang="EN-GB">)
of the polynomial:<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB">Order of coefficients of</span><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas"> +fwd_u and +fwd_v:<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas"> | x^0 | x^1 | x^2 | x^3<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^0 | 1 | 2 | 4 | 7
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^1 | 3 | 5 | 8 | -
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^2 | 6 | 9 | - | -
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:35.4pt"><span lang="EN-GB" style="font-size:10.0pt;font-family:Consolas">y^3 | 10 | - | - | -
<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB">Regards, Jochem<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="EN-GB"><o:p> </o:p></span></p>
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