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:

(1)\[ \begin{align}\begin{aligned}\Delta X = \sum_{i,j} u_{i,j} U^i V^j\\\Delta Y = \sum_{i,j} v_{i,j} U^i V^j\end{aligned}\end{align} \]

where

(2)\[ \begin{align}\begin{aligned}U = X_{in} - X_{origin}\\V = Y_{in} - Y_{origin}\end{aligned}\end{align} \]

and \(u_{i,j}\) and \(v_{i,j}\) are coefficients that make up the polynomial.

The final coordinates are determined as

(3)\[ \begin{align}\begin{aligned}X_{out} = X_{in} + \Delta X\\Y_{out} = Y_{in} + \Delta Y\end{aligned}\end{align} \]

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:

(4)\[\Delta X + i \Delta Y = \sum_{j=1}^n (c_{2j-1} + i c_{2j}) (U + i V)^j\]

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+06,9.9999971567e-01,1.5372750011e-10,5.9300860915e-15,2.2609497633e-19,4.3188227445e-05,2.8225130416e-10,7.8740007114e-16,-1.7453997279e-19,1.6877465415e-10,-1.1234649773e-14,-1.7042333358e-18,-7.9303467953e-15,-5.2906832535e-19,3.9984284847e-19
+fwd_u=8.7760574982e+05,9.9999752475e-01,2.8817299305e-10,5.5641310680e-15,-1.5544700949e-18,-4.1357045890e-05,4.2106213519e-11,2.8525551629e-14,-1.9107771273e-18,3.3615590093e-10,2.4380247154e-14,-2.0241230315e-18,1.2429019719e-15,5.3886155968e-19,-1.0167505000e-18
+inv_v=6.1258103208e+06,1.0000002826e+00,-1.5372762184e-10,-5.9304261011e-15,-2.2612705361e-19,-4.3188331419e-05,-2.8225549995e-10,-7.8529116371e-16,1.7476576773e-19,-1.6875687989e-10,1.1236475299e-14,1.7042518057e-18,7.9300735257e-15,5.2881862699e-19,-3.9990736798e-19
+inv_u=8.7760527928e+05,1.0000024735e+00,-2.8817540032e-10,-5.5627059451e-15,1.5543637570e-18,4.1357152105e-05,-4.2114813612e-11,-2.8523713454e-14,1.9109017837e-18,-3.3616407783e-10,-2.4382678126e-14,2.0245020199e-18,-1.2441377565e-15,-5.3885232238e-19,1.0167203661e-18

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+05,9.99378966275206e-01,-2.82153291753490e-02,-2.27089979140026e-10,-1.77019590701470e-09,1.08522286274070e-14,2.11430298751604e-15
+inv_c=6.13342118787027e+06,4.94690181709311e+05,9.99824464710368e-01,2.82279070814774e-02,7.66123542220864e-11,1.78425334628927e-09,-1.05584823306400e-14,-3.32554258683744e-15

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.

+ellps=<value>

See proj -le for a list of available ellipsoids.

Defaults to “GRS80”.

+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 comma-separated list of coefficients. The number of coefficients is governed by the degree, \(d\), of the polynomial:

\[N = \frac{(d + 1)(d + 2)}{2}\]
+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).

+inv_u=<u_11,u_12,...,u_ij,..,u_mn>

Coefficients for the inverse transformation i.e. latitude to northing as described in (1).

+inv_v=<v_11,v_12,...,v_ij,..,v_mn>

Coefficients for the inverse 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 comma-separated list of coefficients. The number of coefficients is governed by the degree, \(d\), of the polynomial:

\[N = 2d + 2\]
+fwd_c=<c_1,c_2,...,c_N>

Coefficients for the complex forward transformation as described in (4).

+inv_c=<c_1,c_2,...,c_N>

Coefficients for the complex inverse transformation as described in (4).

Optional

+range=<value>

Radius of the region of validity.

+uneg

Express latitude as southing. Only applies for complex polynomials.

+vneg

Express longitude as westing. Only applies for complex polynomials.

Further reading

  1. Wikipedia