Horner polynomial evaluation¶
New in version 5.0.0.
Alias  horner 
Domain  2D and 3D 
Input type  Geodetic and projected coordinates 
Output type  Geodetic and projected coordinates 
The Horner polynomial evaluation scheme is used for transformations between reference frames where one or both are inhomogenous or internally distorted. This will typically be reference frames created before the introduction of space geodetic techniques such as GPS.
Horner polynomials, or Multiple Regression Equations as they are also known as, have their strength in being able to create complicated mappings between coordinate reference frames while still being lightweight in both computational cost and disk space used.
PROJ implements two versions of the Horner evaluation scheme: Real and complex polynomial evaluation. Below both are briefly described. For more details consult [Ruffhead2016] and [IOGP2018].
The polynomial evaluation in real number space is defined by the following equations:
where
and \(u_{i,j}\) and \(v_{i,j}\) are coefficients that make up the polynomial.
The final coordinates are determined as
The inverse transform is the same as the above but requires a different set of coefficients.
Evaluation of the complex polynomials are defined by the following equations:
Where \(n\) is the degree of the polynomial. \(U\) and \(V\) are defined as in (2) and the resulting coordinates are again determined by (3).
Examples¶
Mapping between Danish TC32 and ETRS89/UTM zone 32 using polynomials in real number space:
+proj=horner
+ellps=intl
+range=500000
+fwd_origin=877605.269066,6125810.306769
+inv_origin=877605.760036,6125811.281773
+deg=4
+fwd_v=6.1258112678e+06,9.9999971567e01,1.5372750011e10,5.9300860915e15,2.2609497633e19,4.3188227445e05,2.8225130416e10,7.8740007114e16,1.7453997279e19,1.6877465415e10,1.1234649773e14,1.7042333358e18,7.9303467953e15,5.2906832535e19,3.9984284847e19
+fwd_u=8.7760574982e+05,9.9999752475e01,2.8817299305e10,5.5641310680e15,1.5544700949e18,4.1357045890e05,4.2106213519e11,2.8525551629e14,1.9107771273e18,3.3615590093e10,2.4380247154e14,2.0241230315e18,1.2429019719e15,5.3886155968e19,1.0167505000e18
+inv_v=6.1258103208e+06,1.0000002826e+00,1.5372762184e10,5.9304261011e15,2.2612705361e19,4.3188331419e05,2.8225549995e10,7.8529116371e16,1.7476576773e19,1.6875687989e10,1.1236475299e14,1.7042518057e18,7.9300735257e15,5.2881862699e19,3.9990736798e19
+inv_u=8.7760527928e+05,1.0000024735e+00,2.8817540032e10,5.5627059451e15,1.5543637570e18,4.1357152105e05,4.2114813612e11,2.8523713454e14,1.9109017837e18,3.3616407783e10,2.4382678126e14,2.0245020199e18,1.2441377565e15,5.3885232238e19,1.0167203661e18
Mapping between Danish System Storebælt and ETRS89/UTM zone 32 using complex polynomials:
+proj=horner
+ellps=intl
+range=500000
+fwd_origin=4.94690026817276e+05,6.13342113183056e+06
+inv_origin=6.19480258923588e+05,6.13258568148837e+06
+deg=3
+fwd_c=6.13258562111350e+06,6.19480105709997e+05,9.99378966275206e01,2.82153291753490e02,2.27089979140026e10,1.77019590701470e09,1.08522286274070e14,2.11430298751604e15
+inv_c=6.13342118787027e+06,4.94690181709311e+05,9.99824464710368e01,2.82279070814774e02,7.66123542220864e11,1.78425334628927e09,1.05584823306400e14,3.32554258683744e15
Parameters¶
Setting up Horner polynomials requires many coefficients being explicitly written, even for polynomials of low degree. For this reason it is recommended to store the polynomial definitions in an init file for easier writing and reuse.
Required¶
Below is a list of required parameters that can be set for the Horner polynomial transformation. As stated above, the transformation takes to forms, either using real or complex polynomials. These are divided into separate sections below. Parameters from the two sections are mutually exclusive, that is parameters describing real and complex polynomials can’t be mixed.

+deg
=<value>
¶ Degree of polynomial

+fwd_origin
=<northing,easting>
¶ Coordinate of origin for the forward mapping

+inv_origin
=<northing,easting>
¶ Coordinate of origin for the inverse mapping
Real polynomials¶
The following parameters has to be set if the transformation consists of polynomials in real space. Each parameter takes a commaseparated list of coefficients. The number of coefficients is governed by the degree, \(d\), of the polynomial:

+fwd_u
=<u_11,u_12,...,u_ij,..,u_mn>
¶ Coefficients for the forward transformation i.e. latitude to northing as described in (1).

+fwd_v
=<v_11,v_12,...,v_ij,..,v_mn>
¶ Coefficients for the forward transformation i.e. longitude to easting as described in (1).
Complex polynomials¶
The following parameters has to be set if the transformation consists of polynomials in complex space. Each parameter takes a commaseparated list of coefficients. The number of coefficients is governed by the degree, \(d\), of the polynomial: