.. role:: blue
   :class: blue

MbNurbs3D NURBS Curve
=====================
.. _MbNurbs3D_NURВS-Curve_:

.. kernel version:

The :blue:`MbNurbs3D` class is declared in the *cur\_nurbs3d.h* file. 

The NURBS curve takes its name from the initials of the words NonUniform
Rational B-Spline. The curve is derived from the :blue:`MbPolyCurve3D`
curve. The curve is defined by the set of control points
SArray\<:ref:`MbCartPoint3D <MbCartPoint3D_Radius_Vector_of_Point_in_3D_Space_>`\> **pointList**, the set of weights of the control
points *weights*, the knot vector *knots*, the spline order
*degree*, the curve form parameter *form*, and the curve
periodicity flag *closed*. The curve also has some additional data
that are optional and used to speed up the execution of its methods. 

The curve is constructed based on B-splines. The knot vector *knots*
is a nondecreasing sequence of real numbers and defines the domain of
the curve parameter and the curve shape. The curve form parameter *form*
generally takes the *ncf_Unspecified* value; in special cases it stores
information about the original curve from which the NURBS copy was obtained.
The NURBS curve order *degree* is equal to the order of divided differences
used to compute the B-splines. Let the knot vector contain *knotsCount*
elements and the set of control points contain *pointsCount*
elements. For a non-periodic NURBS curve, the numbers of elements in the
sets are related by
*knotsCount* = *pointsCount* + *degree*. For a periodic
NURBS curve, the numbers of elements in the sets are related by
*knotsCount* = *pointsCount* + 2 *degree* – 1. 

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 the vector function

.. math::

   r(t) = \frac{\sum\limits_{j = 0}^{\mathit{pointsCount} - 1}
                {N_{j^{\mathit{degree}}}(t)\, \mathit{weight}[j]\, \mathbf{pointList}[j]}}
              {\sum\limits_{j = 0}^{\mathit{pointsCount} - 1}
                {N_{j^{\mathit{degree}}}(t)\, \mathit{weight}[j]}},

where
:math:`N_{j}^{\mathit{degree}}(t)` are B-splines of order *degree* for the *j*-th control
point **pointList**\[*j*\]. 

The NURBS curve is shown in Fig. 1: 

.. figure:: /MODELER/CURVES/pictures/O.4.6.1.png
   :width: 12cm
   :align: center

   Fig. 1

The curve parameter *t* takes values on the interval
*tmin* ≤ *t* ≤ *tmax*, where
*tmin* = *knots* \[*degree* – 1\],
*tmax* = *knots* \[*knotsCount* – *degree*\]. The curve
may be periodic. A periodic NURBS curve is shown in Fig. 2: 

.. figure:: /MODELER/CURVES/pictures/O.4.6.2.png
   :width: 12cm
   :align: center

   Fig. 2

The NURBS curve shape depends on the control point locations, the control
point weights, and the knot vector values. In the general case, a NURBS
curve does not pass through the set of points **pointList**\[i\],
*i* = 0,..., **pointsCount** – 1. For an open NURBS curve to pass
through the end control points, the first *degree* elements and the last
*degree* elements of the knot vector *knots* must coincide. The larger
the weight of a control point, the closer the curve passes to this point,
all other conditions being equal. 

Each curve can construct its own NURBS copy using the virtual method
:blue:`NurbsCurve`\( const MbNurbsParameters & *tParameters* ).