Grapher
=======

The continuous function within the interval.

.. math::

   r: t\longmapsto r(t), t \in \left[ t_{0}, t_{n} \right], t \in \mathbb{D}, n \in \mathbb{N}

.. image:: _static/function_graph.svg
   :height: 350pt

These two entities are being used to construct a planar graph of consecutive vectors (Banach space).

.. math::
   t_{n}-t_{n-1}\text{: Distance between two elements in r(t)}
.. math::
   \alpha (t):\text{Angle between two elements in r(t)}

Building the vector graph starting at origin.

.. math::
   \mathbf{v_{0}} = \begin{pmatrix}
   0 \\
   0
   \end{pmatrix}
   \quad
   \mathbf{v_{n}} = \mathbf{v_{n-1}} + \mathbf{v}(t_{n})

.. image:: _static/vector_graph.svg
   :height: 350pt
   
The vector for each element of r(t) built from the rotation of all preceding element-vectors, its own angle and distance.

.. math::
   \mathbf{v}(t) = M(\int\limits_{t_{0}}^{t_{n}}\alpha(t))\mathrm{d}t \cdot \begin{pmatrix} 
   \mathrm{d}t \\
   0
   \end{pmatrix}
   \approx M(\displaystyle\sum_{t_{1}}^{t_{n}}\alpha(t)) \cdot \begin{pmatrix}
   t_{n}-t_{n-1} \\
   0
   \end{pmatrix}, n \neq 0
.. math::
   \alpha(t) = \arctan \frac{a * r'(t)}{t} \approx \arctan \frac{a(r(t_{n})-r(t_{n-1}))}{t_{n}-t_{n-1}}, n \neq 0
.. math::
   M(\alpha):\text{2x2 rotation matrix of angle }\alpha
.. math::
   M(\alpha) = \begin{pmatrix}
   \cos \alpha & -\sin \alpha \\
   \sin \alpha & \cos \alpha
   \end{pmatrix}

In the example given below, for simplicity r(t) is not time-variant. Otherwise step-length could vary.

.. math::
   \Delta q=\Delta t

.. image:: _static/vector_construction.svg
   :height: 400pt

Within the interval satisfying the condition,

.. math::
   q \in \left[ t_{x}, t_{y} \right], k * \Delta q=t_{n}-t_{n-1}, k \in \mathbb{N}, k \neq 0

the scaling factor using the arithmetic mean of changes in value of r(t) is.

.. math::
   a = \frac{1}{\frac{\int\limits_{t_{x}}^{t_{y}}\left|r'(q)\right|\mathrm{d}q}{\frac{t_{y}-t_{x}}{t}}} \approx \frac{1}{\frac{\displaystyle\sum_{t_{x}}^{t_{y}}\left|r(t_{y})-r(t_{y-1})\right|}{\frac{t_{y}-t_{x}}{t_{n}-t_{n-1}}}}, n \neq 0
   \quad
   a \in \mathbb{R}

A different scaling factor can be chosen. The criteria for choosing a scaling factor is whether the shortest impulse in the non time-variant function r(q) forms a distinguishable shape in the planar graph.

This would be something similar to the above vector graph using a different factor.

.. image:: _static/vector_graph2.svg
   :height: 350pt

And something similar using an even bigger factor.

.. image:: _static/vector_graph3.svg
   :height: 300pt

Once the vector graph has been calculated, it can be used as input for the Analyzer.