.. role:: blue
   :class: blue


MbCubicSplineFunction Cubic Spline Function
===========================================
.. _MbCubicSplineFunction_Cubic_Spline_Function_:


.. kernel version: 

The :blue:`MbCubicSplineFunction` class is declared in the *cur_cubic_spline_function.h* file.

The cubic spline function :blue:`MbCubicSplineFunction` is defined by the set of
control points *valueList*, the set of second derivatives of the
function at the control points *secondList*, the set of parameter values
of the function at the control points *tList*, and the flag of
periodicity of the function *closed*. The function also has some
additional data that are optional and are used to speed up the work of
the function methods.

For the parameter value
*tList*\[*i*\], *i* = 0, 1, ..., *splinesCount*, where
*splinesCount* = *tList*.MaxIndex(), the cubic spline function passes
through the control point *valueList*\[*i*\] and has the second
derivative *secondList*\[*i*\] at this point.

When evaluating the function, first, by the value of the curve parameter
*t*, the index *i* of the working segment is determined from the
condition *tList*\[*i*\] ≤ *t* ≤ *tList*\[*i* + 1\].
Then the function is evaluated for the found working segment using its
local parameter *w*, which is defined by *tList*\[*i*\]
and *tList*\[*i* + 1\].

In the double :blue:`Value`\( double & *t* ) method, the following
function is used


.. math::

   r(t) = (1 - w) \cdot \mathit{\text{valueList}}[i] +
          w \cdot \mathit{\text{valueList}}[i + 1] +
           \left( (- 2 w + 3 w^{2} - w^{3}) \cdot \mathit{\text{valueList}}[i] +
                  (- w + w^{3}) \cdot \mathit{\text{valueList}}[i + 1] \right)
            \frac{(\mathit{\text{tList}}[i + 1] - \mathit{\text{tList}}[i])^{2}}{6},

where


.. math::

   w = \frac{ t - \mathit{\text{tList}}[i] }
            { \mathit{\text{tList}}[i + 1] - \mathit{\text{tList}}[i] }


is the local parameter of the working segment tList[i] ≤ t ≤ tList[i + 1].

The cubic spline function is shown in Fig. 1:

.. figure:: /MODELER/SPECIAL_OBJECTS/pictures/O.6.5.1.png
   :width: 15cm
   :align: center

   Fig. 1


The domain of the function parameter lies within
*tmin* ≤ *t* ≤ *tmax*, where *tmin* = *tList*\[0\],
*tmax* = *tList*\[*splinesCount*\]. The function can be
periodic.


The shape of the function depends on the positions of the control points
and on the *tList* set of parameter values at the control points. When
the function is constructed by control points, the derivatives
*secondList*\[*i*\] are computed under the assumption of linear
variation of the second derivatives between the control points.