.. role:: blue
   :class: blue


MbCubicFunction Cubic Hermite Function
======================================
.. _MbCubicFunction_Cubic_Hermite_Function_:

.. kernel version: 

The :blue:`MbCubicFunction` class is declared in the *cur_cubic_function.h* file.

The Cubic Hermite function :blue:`MbCubicFunction` is defined by the set of
control points *valueList*, the set of function derivatives at the
control points *firctList*, 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 Hermite function passes
through the control point *valueList*\[*i*\] and has the derivative
*firctList*\[*i*\] at this point. The function is constructed on the
basis of *splinesCount* third-degree Hermite splines, which smoothly
join each other. Each cubic Hermite spline describes a segment of the
function between two neighboring control points. Each cubic Hermite
spline is defined by two end points and two derivatives of the curve at
these points.

When evaluating the function, first, by the value of the curve parameter
*t*, the index *i* of the working segment (the index of the cubic
Hermite spline) 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::

   f(t) = (1 - 3 w^{2} + 2 w^{3}) \, valueList[i] +
          (3 w^{2} + 2 w^{3}) \, valueList[i + 1] + \\
          + \left( (w - 2 w^{2} + w^{3}) \, valueList[i] +
                  (- w^{2} + w^{3}) \, valueList[i + 1] \right)
            ( tList[i + 1] - tList[i] ),

where

.. math::

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

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

The Cubic Hermite function is shown in Fig. 1:

.. figure:: /MODELER/SPECIAL_OBJECTS/pictures/O.6.4.1.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,
on the function derivatives at the control points, and on the *tList* set
of parameter values at the control points. When the function is
constructed only by control points, the derivatives *firctList*\[*i*\]
are computed by constructing a parabola that passes through three
neighboring points *valueList*\[*i* -- 1\],
*valueList*\[*i*\], *valueList*\[*i* + 1\] at the
corresponding parameter values *tList*\[*i* -- 1\],
*tList*\[*i*\], *tList*\[*i* + 1\], and by evaluating
the derivative of this parabola at the middle point.