Description of document versions
====================================


.. csv-table::

    "**Version** **Aster**", "**Author (s)**
    **Organization (s)**", "**Description of changes**"
    "8.4", "J. ANGLES
    EDF -R&D/ AMA ", "Initial text"


The various situations are summarized in [:ref:`Tableau A1-1 <Tableau A1-1>`]. In [:ref:`Tableau A1-1 <Tableau A1-1>`], "0" and "1" respectively mean that there are no points and that there is at least one point in the designated sectors.


.. csv-table::

    "**Sector 1**", "**Sector 3**", "**Sector 2**", "**Sector 4**", "**Area of Projection**"
    "0", "0", "0", "0", "Impossible case."
    "0", "0", "0", "0", "1", "Impossible case."
    "0", "0", "1", "0", "Impossible case."
    "0", "0", "1", "1", "1", "Axis 1."
    "0", "1", "0", "0", "0", "Impossible case."
    "0", "1", "0", "1", "Using the selection procedure."
    "0", "1", "1", "0", "0", "Using the selection procedure."
    "0", "1", "1", "1", "1", "Axis 1."
    "1", "0", "0", "0", "0", "Impossible case."
    "1", "0", "0", "1", "Using the selection procedure."
    "1", "0", "1", "0", "0", "Using the selection procedure."
    "1", "0", "1", "1", "1", "Axis 1."
    "1", "1", "0", "0", "0", "Axis 2."
    "1", "1", "0", "1", "1", "Axis 2."
    "1", "1", "1", "0", "0", "Axis 2."
    "1", "1", "1", "1", "1", "Using the selection procedure."


**Table A1-1: Summary of situations**

Impossible cases result from the way in which the framework and the sectors are constructed. This construction makes it impossible for points to be present in any or only one sector.

The projection of any point on the second axis is quickly described in this appendix. Starting from a point

.. image:: images/1000013A000027B000002AFF8B43B1A54DD1B4D8.svg
    :width: 15
    :height: 17


.. _RefImage_1000013A000027B000002AFF8B43B1A54DD1B4D8.svg:

Whatever is known, we calculate the coordinates of a point

.. image:: images/1000019200002E4D00002AFF94907D22802342CB.svg
    :width: 15
    :height: 17


.. _RefImage_1000019200002E4D00002AFF94907D22802342CB.svg:

such as:


.. image:: images/100002D8000069D500001EB2EC5F1AA2B25CADA8.svg
    :width: 15
    :height: 17


.. _RefImage_100002D8000069D500001EB2EC5F1AA2B25CADA8.svg:

After simplification comes the relationship:

Where a value of

.. image:: images/100001EA00003ED700003839E07CCDBA237D4B56.svg
    :width: 15
    :height: 17


.. _RefImage_100001EA00003ED700003839E07CCDBA237D4B56.svg:

different from

.. image:: images/1000019200003839000038399C7EC83CFB1C8805.svg
    :width: 15
    :height: 17


.. _RefImage_1000019200003839000038399C7EC83CFB1C8805.svg:

Give us

.. image:: images/100001EA000034EB00003839533ED9CB4DE90AD3.svg
    :width: 15
    :height: 17


.. _RefImage_100001EA000034EB00003839533ED9CB4DE90AD3.svg:

.

In the plan

.. image:: images/100001C200005FE9000034EB48175096F7C33361.svg
    :width: 15
    :height: 17


.. _RefImage_100001C200005FE9000034EB48175096F7C33361.svg:

the second axis and the segment are affine lines respectively described by

.. image:: images/1000023A000069D500001FBB3BE93F50C9B26B62.svg
    :width: 15
    :height: 17


.. _RefImage_1000023A000069D500001FBB3BE93F50C9B26B62.svg:

and

.. image:: images/1000023A000069D500001C87A6C962D3FCE3EB01.svg
    :width: 15
    :height: 17


.. _RefImage_1000023A000069D500001C87A6C962D3FCE3EB01.svg:

, so to know the coordinates of the point projected on the second axis

.. image:: images/100001920000319C00003ED7E86C827C8FFAE6D2.svg
    :width: 15
    :height: 17


.. _RefImage_100001920000319C00003ED7E86C827C8FFAE6D2.svg:

we solve the equation:


.. image:: images/1000028A000069D5000013A330DD0E86D2C1ACCC.svg
    :width: 15
    :height: 17


.. _RefImage_1000028A000069D5000013A330DD0E86D2C1ACCC.svg:

where

.. csv-table::

    ":math:`{a}_{s}=\frac{({V}_{M}-{V}_{O})}{({U}_{M}-{U}_{O})}`,", ":math:`{b}_{s}=\frac{({U}_{M}{V}_{O}-{U}_{O}{V}_{M})}{({U}_{M}-{U}_{O})}`,"
    ":math:`{a}_{P}=\frac{({V}_{{P}^{\text{'}}}-{V}_{P})}{({U}_{{P}^{\text{'}}}-{U}_{P})}`,", "."


We get:

+------------------------------------------------------------------------------------------------------------------+
|:math:`{U}_{{P}_{s}}=\frac{{b}_{P}-{b}_{s}}{{a}_{s}-{a}_{P}}`,                                                    |
+------------------------------------------------------------------------------------------------------------------+
|                                                                                                                  |
+ .. image:: images/1000034E000069D5000029A7BD23CE5EAC4CAAB3.svg                                                   +
|    :width: 15                                                                                                    |
+    :height: 17                                                                                                   +
|                                                                                                                  |
+.                                                                                                                 +
|                                                                                                                  |
+------------------------------------------------------------------------------------------------------------------+