.. role:: blue
   :class: blue

Polyline MbPolyline3D
=====================
.. _MbPolyline3D_Polyline_:

.. kernel version: 

The :blue:`MbPolyline3D` class is declared in the *cur\_polyline3d.h* file. 

The polyline is derived from the :blue:`MbPolyCurve3D` curve. The :blue:`MbPolyline3D`
polyline is defined by the number of segments *segmentsCount*, the set of
control points SArray\<:ref:`MbCartPoint3D <MbCartPoint3D_Radius_Vector_of_Point_in_3D_Space_>`\> **pointList**
and the curve periodicity flag *closed*. 

The curve passes through the set of points **pointList**\[*i*\],
*i* = 0,..., *segmentsCount*, at parameter values *t* = 0,..., *segmentsCount*.
If *closed* = true, then the curve contains a segment connecting the last point of the set
**pointList**\[*segmentsCount* - 1\] with the first point **pointList**\[0\].
The curve parameter *t* takes values on the interval 0 ≤ *t* ≤ *segmentsCount*. 

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

.. math::

   \mathbf{r}(t) = \mathbf{pointList}[i] \cdot (1 - w) + \mathbf{pointList}[i + 1] \cdot w,

where

.. math::

   w = \frac{t - t_{i}}{t_{i+1} - t_{i}},

and *t*\ :sub:`i` ≤ *t* ≤ *t*\ :sub:`i+1`.
The polyline is the simplest curve constructed from a set of points. It
consists of segments that consecutively connect the control points. 

The polyline is shown in the figure:

.. figure:: /MODELER/CURVES/pictures/O.4.5.1.png
   :width: 12cm
   :align: center

The curve may be periodic. The period of a periodic curve is equal to
*segmentsCount*. The derivatives of the curve at the control points (for
integer parameter values) lose continuity in length and direction. The derivatives of the curve at the control points have a specific
direction and length. The polyline has a number of useful properties: it
requires a minimum of computations, and the projection of a polyline is also
a polyline.