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Add invertible Horner polynomials #3133

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Add invertible Horner polynomials #3133

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3159792
Add inversible horner real polynomials
georgeouzou Mar 23, 2022
f14feeb
Fix windows compilation error
georgeouzou Mar 24, 2022
12c3377
Handle some inversible horner corner cases
georgeouzou Mar 24, 2022
04baa72
Add documentation for iterative real polynomial inversion
georgeouzou Mar 25, 2022
3dbe701
Address docs review comments
georgeouzou Mar 26, 2022
98e4a89
Add inversible horner complex polynomials
georgeouzou Mar 26, 2022
16aab79
Optimize iterative inversion of complex horner
georgeouzou Mar 26, 2022
1b556a8
Fix sanitizer errors
georgeouzou Mar 26, 2022
d75519b
Apply review comments
georgeouzou Mar 31, 2022
a02f426
Add doc note on reversible polynomials
georgeouzou Apr 16, 2022
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126 changes: 104 additions & 22 deletions docs/source/operations/transformations/horner.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -5,6 +5,7 @@ Horner polynomial evaluation
================================================================================

.. versionadded:: 5.0.0
.. versionchanged:: 9.1.0 Iterative polynormal inversion

+-----------------+-------------------------------------------------------------------+
| **Alias** | horner |
Expand Down Expand Up @@ -61,7 +62,63 @@ The final coordinates are determined as
Y_{out} = Y_{in} + \Delta Y

The inverse transform is the same as the above but requires a different set of
coefficients.
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 :eq:`real_poly` as:

.. math::
\begin{bmatrix}
\Delta X \\
\Delta Y \\
\end{bmatrix} =
\begin{bmatrix}
u_{0,0} \\
v_{0,0} \\
\end{bmatrix} +
\begin{bmatrix}
u_{0,1} + u_{0,2} U + ... & u_{1,0} + u_{1,1} U + u_{2,0} V + ... \\
v_{1,0} + v_{1,1} V + v_{2,0} U + ... & v_{0,1} + v_{0,2} V \\
\end{bmatrix}
\begin{bmatrix}
U \\
V \\
\end{bmatrix} \\

.. math::
\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} \\

.. math::
\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}

We can iteratively solve with initial values of :math:`U = 0` and :math:`V = 0` and find :math:`U` and :math:`V`.

.. note::
This iterative inverse transformation is a more general solution to *reversible polynormials of degree n* as presented in :cite:`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:

Expand All @@ -72,7 +129,7 @@ Evaluation of the complex polynomials are defined by the following equations:

Where :math:`n` is the degree of the polynomial. :math:`U` and :math:`V` are
defined as in :eq:`UV` and the resulting coordinates are again determined
by :eq:`xy_out`.
by :eq:`xy_out`. Complex polynomials can be solved iteratively similar to real polynomials.

Examples
################################################################################
Expand Down Expand Up @@ -131,10 +188,6 @@ describing real and complex polynomials can't be mixed.

Coordinate of origin for the forward mapping

.. option:: +inv_origin=<northing,easting>

Coordinate of origin for the inverse mapping

Real polynomials
..............................................................................

Expand All @@ -157,18 +210,6 @@ of the polynomial:
Coefficients for the forward transformation i.e. longitude to easting
as described in :eq:`real_poly`.

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

Coefficients for the inverse transformation i.e. latitude to northing
as described in :eq:`real_poly`.

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

Coefficients for the inverse transformation i.e. longitude to easting
as described in :eq:`real_poly`.



Complex polynomials
..............................................................................

Expand All @@ -186,13 +227,43 @@ of the polynomial:
Coefficients for the complex forward transformation
as described in :eq:`complex_poly`.

Optional
-------------------------------------------------------------------------------

.. option:: +inv_origin=<northing,easting>

.. versionchanged:: 9.1.0

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

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

.. versionchanged:: 9.1.0

Coefficients for the inverse transformation i.e. latitude to northing
as described in :eq:`real_poly`. Only applies for real polynomials.
Without this option iterative polynomial evaluation is used for
the inverse tranformation.

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

.. versionchanged:: 9.1.0

Coefficients for the inverse transformation i.e. longitude to easting
as described in :eq:`real_poly`. Only applies for real polynomials.
Without this option iterative polynomial evaluation is used for
the inverse tranformation.

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

Coefficients for the complex inverse transformation
as described in :eq:`complex_poly`.
.. versionchanged:: 9.1.0

Optional
-------------------------------------------------------------------------------
Coefficients for the complex inverse transformation
as described in :eq:`complex_poly`. Only applies for complex polynomials.
Without this option iterative polynomial evaluation is used for
the inverse tranformation.

.. option:: +range=<value>

Expand All @@ -206,6 +277,17 @@ Optional

Express longitude as westing. Only applies for complex polynomials.

.. option:: +inv_tolerance=<value>

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.. versionadded:: 9.1.0

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 :math:`U` and :math:`V` of :eq:`UV`

*Defaults to 0.001.*

Further reading
################################################################################
Expand Down
11 changes: 11 additions & 0 deletions docs/source/references.bib
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Expand Up @@ -169,6 +169,17 @@ @TechReport{IOGP2018
Url = {https://www.iogp.org/bookstore/product/coordinate-conversions-and-transformation-including-formulas/}
}

@TechReport{IOGP2019,
Title = {Geomatics Guidance Note 7, part 2: Coordinate Conversions \& Transformations including Formulas},
Author = {IOGP},
Institution = {International Association For Oil And Gas Producers},
Year = {2019},
Number = {373-7-2},
Type = {IOGP Publication},

Url = {https://www.iogp.org/wp-content/uploads/2019/09/373-07-02.pdf}
}

@TechReport{ISO19111,
Title = {{Geographic information -- Referencing by coordinates}},
Author = {ISO},
Expand Down
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