.. role:: blue
   :class: blue


Cubic Spline MbCubicSpline3D
============================
.. _MbCubicSpline3D_Cubic_Spline_:

.. kernel version: 

The :blue:`MbCubicSpline3D` class is declared in the **cur_cubic_spline3d.h** file.

The cubic spline is a derived curve of MbPolyCurve3D. The curve
is defined by an array of control points
SArray<:ref:`MbCartPoint3D <MbCartPoint3D_Radius_Vector_of_Point_in_3D_Space_>`> **pointList**, an array of the second derivatives
of the curve at the control points
SArray<:ref:`MbVector3D <MbVector3D_Vector_in_Three-Dimensional_Space_>`> **vectorList**, an array of the
curve parameter values at the control points *tList*, the maximum
index value of the parameter array *splinesCount*, and the flag
of curve periodicity *closed*. The curve also contains some additional data
that is optional and is used to speed up the curve methods.

For the cubic spline, at the parameter value *tList*\[*i*\],
*i* = 0, 1, ..., *splinesCount*, the curve passes through the control point
**pointList**\[*i*\] and has the second derivative
**vectorList**\[*i*\] at this point. The curve is constructed so that when
moving from **pointList**\[*i*]\ to
**pointList**\[*i* + 1], the second derivative of the radius vector
of the curve changes linearly from **vectorList**\[*i*\] to
**vectorList**\[*i* + 1].

When computing the radius vector of a point on the composite curve, first,
by the curve parameter value *t* the index *i* of the active
segment is determined from the condition
*tList*\[*i*] ≤ *t* ≤ *tList*\[*i* + 1].
The radius vector of the curve is then computed from
**pointList**\[*i*], **pointList**\[*i* + 1],
**vectorList**\[*i*], **vectorList**\[*i* + 1]
of the found segment for the local parameter *w*, which is
determined by *tList*\[*i*]
and *tList*\[*i* + 1].

In the :blue:`PointOn`\( double & *t*,
:ref:`MbCartPoint3D <MbCartPoint3D_Radius_Vector_of_Point_in_3D_Space_>` & **r**) method, the radius vector **r** of the curve is defined by the vector function

.. math::

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

where

.. math::

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

and *tList*\[*i*] ≤ *t* ≤ *tList*\[*i* + 1].

A cubic spline constructed through the same control points as
the Hermite composite curve is shown in the figure:

.. figure:: /MODELER/CURVES/pictures/O.4.9.1.png
   :width: 15cm
   :align: center

The curve parameter *t* takes values on the interval
*tmin* ≤ *t* ≤ *tmax*, where *tmin* = *tList*\[0],
*tmax* = *tList*\[*splinesCount*]. The curve can be
periodic.

The shape of the curve depends on the arrangement of the control points and on the
*tList* array of parameter values at the control points. When the curve is
constructed only from the control points, the curve parameter values at the
control points *tList*\[*i*],
*i* = 0, 1, ..., *splinesCount*, vary proportionally to the
distance between the points, and the second derivatives
**vectorList**\[*i*],
*i* = 1, 2, ..., *splinesCount* – 1 are computed by solving
a system of equations.