+

+ +

+

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 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 +[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>¶

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
+

+ +

++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¶

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++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.
+

+ +

+