# Horner polynomial evaluation¶

Changed in version 9.1.0: Iterative polynormal inversion

 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 inhomogeneous 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 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 order of coefficients $$u_{i,j}$$ is (example for degree 3):

 V⁰ V¹ V² V³ U⁰ 1st 5th 8rd 10th U¹ 2nd 6th 9th – U² 3rd 7th – – U³ 4th – – –

The order of coefficients $$v_{i,j}$$ is (example for degree 3):

 V⁰ V¹ V² V³ U⁰ 1st 2nd 3rd 4th U¹ 5th 6th 7th – U² 8th 9th – – U³ 10th – – –

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. If only the forward set of coefficients and origin is known the inverse transform can be done by iteratively solving a system of equations. By writing (1) as:

$\begin{split}\begin{bmatrix} \Delta X \\ \Delta Y \\ \end{bmatrix} = \begin{bmatrix} u_{0,0} \\ v_{0,0} \\ \end{bmatrix} + \begin{bmatrix} u_{1,0} + u_{2,0} U + ... & u_{0,1} + u_{1,1} U + u_{0,2} V + ... \\ v_{0,1} + v_{1,1} V + v_{0,2} U + ... & v_{1,0} + v_{2,0} V \\ \end{bmatrix} \begin{bmatrix} U \\ V \\ \end{bmatrix} \\\end{split}$
$\begin{split}\begin{bmatrix} \Delta X \\ \Delta Y \\ \end{bmatrix} = \begin{bmatrix} u_{0,0} \\ v_{0,0} \\ \end{bmatrix} + \begin{bmatrix} MA & MB \\ MC & MD \\ \end{bmatrix} \begin{bmatrix} U \\ V \\ \end{bmatrix} \\\end{split}$
$\begin{split}\begin{bmatrix} U \\ V \\ \end{bmatrix} = \begin{bmatrix} MA & MB \\ MC & MD \\ \end{bmatrix}^{-1} \begin{bmatrix} \Delta X - u_{0,0} \\ \Delta Y - v_{0,0} \\ \end{bmatrix}\end{split}$

We can iteratively solve with initial values of $$U = 0$$ and $$V = 0$$ and find $$U$$ and $$V$$.

Note

This iterative inverse transformation is a more general solution to reversible polynormials of degree n as presented in [IOGP2019]. These can provide a satisfactory solution in a single step when certain conditions are met.

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). Complex polynomials can be solved iteratively similar to real polynomials.

## 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>

The name of a built-in ellipsoid definition.

See Ellipsoids for more information, or execute proj -le for a list of built-in ellipsoid names.

Defaults to "GRS80".

+deg=<value>

Degree of polynomial

+fwd_origin=<northing,easting>

Coordinate of origin for the forward 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_00,u_10,...,u_ij,..,u_nn>

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

+fwd_v=<v_00,v_01,...,v_ij,..,v_nn>

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 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).

### Optional¶

+inv_origin=<northing,easting>

Changed in version 9.1.0.

Coordinate of origin for the inverse mapping. Without this option iterative polynomial evaluation is used for the inverse transformation.

+inv_u=<u_00,u_10,...,u_ij,..,u_nn>

Changed in version 9.1.0.

Coefficients for the inverse transformation i.e. latitude to northing as described in (1). Only applies for real polynomials. Without this option iterative polynomial evaluation is used for the inverse transformation.

+inv_v=<v_00,v_01,...,v_ij,..,v_nn>

Changed in version 9.1.0.

Coefficients for the inverse transformation i.e. longitude to easting as described in (1). Only applies for real polynomials. Without this option iterative polynomial evaluation is used for the inverse transformation.

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

Changed in version 9.1.0.

Coefficients for the complex inverse transformation as described in (4). Only applies for complex polynomials. Without this option iterative polynomial evaluation is used for the inverse transformation.

+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.

+inv_tolerance=<value>

Only applies to cases of iterative inversion. The procedure converges to the correct results with each step. Iteration stops when the result differs from the previous calculated result by less than <value>. <value> should be the same units as $$U$$ and $$V$$ of (2)