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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