Choice of projection axes
============================

With regard to the projection of the end of the cission vector we propose two options:


* a projection on an axis,


* a projection on two axes.

The axis for option 1 is determined in the same way as the first axis for option 2. The second axis in option 2 is orthogonal to the first axis in this option.




Projection on an axis
---------------------

We place ourselves in a given normal :math:`\overrightarrow{n}` plane where each point represents the position of the tip of the shear vector at a given time, for more details see reference [:ref:`bib6 <bib6>`]. In this plane we build the smallest frame that contains all the points representing the end of the cission vector at each point in time. The two diagonals of the frame allow us to define two axes: axis 1 corresponds to segment :math:`\overline{\text{AC}}`, and axis 2 corresponds to segment :math:`\overline{\text{DB}}`, cf. [:ref:`Figure 7.1 <Figure 7.1>`].


.. image:: images/Object_564.svg
    :width: 371
    :height: 244


.. _RefImage_Object_564.svg:

**Figure 7.1: Definition of the axes of projection**

In principle, we choose the projection axis from among the axes 1 and 2 because the diagonal of the frame is larger than the long side of the frame, which has the virtue of dilating the projected points a little. On the other hand, the loads that interest us are of thermal origin, which means that the points representing the evolution of the tip of the cission vector, in the planes of normal :math:`\overrightarrow{n}`, are most often aligned on an axis, as we show in figure [:ref:`Figure 6.1-a <Figure 6.1-a>`].

Sectors 1, 2, 3, and 4 are constructed in the same way as in the reference [:ref:`bib6 <bib6>`]. Only the points in these sectors are projected orthogonally onto axes 1 and 2.

We define the projection axis as the axis on which the distance between two projected points is greatest.

For example, on [:ref:`Figure 7.1-a <Figure 7.1-a>`] the projection axis is axis 1 since the length of segment :math:`\overline{{P}_{3}{P}_{4}}` is greater than the length of segment :math:`\overline{{P}_{1}{P}_{2}}`. This definition of the projection axis makes it possible to ensure that the projection axis selected will make it possible to account for the largest projected shear amplitude.


.. _OLE_LINK1:


Depending on the presence or absence of points in sectors 1, 2, 3 and 4, the determination of the projection axis may be immediate; it is then not necessary to implement the selection procedure described above. For more details, the reader can refer to Appendix 1.

A second axis is necessary to distinguish the case where the points representing the tip of the cission vector are aligned on an axis from the case where these points describe a circle.




Construction of the second axis
--------------------------

The second axis of projection is orthogonal to the initial axis of projection and passes through the point :math:`O`.

Since we know the coordinates of the points :math:`A`, :math:`B`,, :math:`C` and :math:`D`, to fully characterize the second axis it suffices to determine the coordinates of a point :math:`M` such as:

:math:`\overrightarrow{\text{DB}}\text{.}\overrightarrow{\text{OM}}\mathrm{=}0` if the initial axis is axis 1,

:math:`\overrightarrow{\text{AC}}\text{.}\overrightarrow{\text{OM}}\mathrm{=}0` if the initial axis is axis 2.