Equations for meridian convergence and partial derivatives reviewed (corrected)

[dtutic]

I have found strange results for meridian convergence in non-conformal projections. As we know from map projection books (with Gauss-Bomford convention for convergence), meridian convergence should be calculated by formula fac->conv = - atan2(fac->der.x_p , fac->der.y_p).

Current formula used in Proj4 is fac->conv = - atan2(fac->der.y_l , fac->der.x_l), which can be useful only for conformal projections (due to Cauchy-Riemann equations), and in fact should be without minus sign, i.e. fac->conv = atan2(fac->der.y_l , fac->der.x_l). For non-conformal projections we now get angle between parallel and x-axis, instead of meridian convergence.

Minus sign currently used in formula comes from fact that fac->der.x_p and fac->der.y_l are calculated with opposite signs.

Corrected equations are:

File: pj_deriv.c

Line 14
instead of:
der->x_l = t.x; der->y_p = t.y; der->x_p = -t.x; der->y_l = -t.y;
should be:
der->x_l = t.x; der->y_p = t.y; der->x_p = t.x; der->y_l = t.y;

Line 19
instead of:
der->x_l += t.x; der->y_p -= t.y; der->x_p += t.x; der->y_l -= t.y;
should be:
der->x_l += t.x; der->y_p -= t.y; der->x_p -= t.x; der->y_l += t.y;

Line 23
instead of:
der->x_l -= t.x; der->y_p -= t.y; der->x_p += t.x; der->y_l += t.y;
should be:
der->x_l -= t.x; der->y_p -= t.y; der->x_p -= t.x; der->y_l -= t.y;

Line 27
instead of:
der->x_l -= t.x; der->y_p += t.y; der->x_p -= t.x; der->y_l += t.y;
should be:
der->x_l -= t.x; der->y_p += t.y; der->x_p += t.x; der->y_l -= t.y;

File: pj_factors.c

Line 72
instead of:
fac->conv = - atan2(fac->der.y_l, fac->der.x_l);
should be:
fac->conv = - atan2(fac->der.x_p, fac->der.y_p);

One can question the practical use of meridian convergence in non-conformal projections, but nevertheless, it would be nice to have it calculated correctly.

[kbevers]

I am not able to comment on the details of this without researching a bit first, but at least it sounds sensible.

I have created a pull request with your changes in #527. I have reformatted the code as well. Please check that the changes are equivalent to your suggestion.