[PROJ] Horner polynomial evaluation

[Lesparre, Jochem]

Dear list members,

The implementation of +proj=horner 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 https://proj.org/en/9.3/operations/transformations/horner.html. The order of the coefficients in the formulas and in the description of the PROJ parameters (+fwd_u and +fwd_u) seem mutually inconsistent.

Problem 1: The matrix formula (after formula 3) seems inconsistent with formula (1) and the actual implementation.


\begin{bmatrix}
     u_{0,1} + u_{0,2} U + ... & u_{1,0} + u_{1,1} U + u_{2,0} V + ... \\
     v_{1,0} + v_{1,1} V + v_{2,0} U + ... & v_{0,1} + v_{0,2} V \\
\end{bmatrix}

should be changed to:

\begin{bmatrix}
     u_{1,0} + u_{2,0} U + ... & u_{0,1} + u_{1,1} U + u_{0,2} V + ... \\
    v_{0,1} + v_{1,1} V + v_{0,2} U + ... & v_{1,0} + v_{2,0} V \\
\end{bmatrix}

Problem 2: The description of the parameters seems inconsistent with formula (1) and the actual implementation.

+fwd_u=<u_11,u_12,...,u_ij,..,u_mn>
and
+fwd_v=<v_11,v_12,...,v_ij,..,v_mn>

should be changed to:

+fwd_u=<u_00,u_10,...,u_ij,..,u_nn>
and
+fwd_v=<v_00,v_01,...,v_ij,..,v_nn>


As a result it was difficult to determine the correct order of the coefficients for +fwd_u and +fwd_v parameters. I think it would be clarifying to add a table for the order of the coefficients in the documentation (example for +deg=3):

Order of coefficients of +fwd_u:
    | x^0 | x^1 | x^2 | x^3
y^0 | 1   | 2   | 3   | 4
y^1 | 5   | 6   | 7   | -
y^2 | 8   | 9   | -   | -
y^3 | 10  | -   | -   | -

Order of coefficients of +fwd_v:
    | x^0 | x^1 | x^2 | x^3
y^0 | 1   | 5   | 8   | 10
y^1 | 2   | 6   | 9   | -
y^2 | 3   | 7   | -   | -
y^3 | 4   | -   | -   | -

Suggestion for a different implementation: 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 +fwd_u and +fwd_v, and it would keep the first coefficients the same when increasing the degree (+deg) of the polynomial:

Order of coefficients of +fwd_u and +fwd_v:
    | x^0 | x^1 | x^2 | x^3
y^0 | 1   | 2   | 4   | 7
y^1 | 3   | 5   | 8   | -
y^2 | 6   | 9   | -   | -
y^3 | 10  | -   | -   | -

Regards, Jochem



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