.. role:: blue
   :class: blue


MbSplineSurface NURBS Surface
=============================
.. _MbSplineSurface_NURBS_Surface_:


.. версия ядра: 1180ХХ


Class :blue:`MbSplineSurface` is declared in the *surf\_spline\_surface.h* file.

The NURBS surface (NonUniform Rational B-Spline surface) :blue:`MbSplineSurface` is defined by control points SArray<:blue:`MbCartPoint3D`> **points**\[*i*]\[*j*], *i* = 0, 1, ..., *vcount* − 1, *j* = 0, 1, ..., *ucount* − 1, conventionally located at the nodes of a rectangular grid with *ucount* columns and *vcount* rows, by the weights of the control points given in the table *weight*\[*i*]\[*j*], by the order of B-splines along the first surface parameter *udegree*, by the order of B-splines along the second surface parameter *vdegree*, by the knot vector along the first parameter *uknots*, by the knot vector along the second parameter *vknots*, and by the flags of the surface periodicity in the first and second parameters *uclosed* and *vclosed*. The surface also has some additional data that is optional and is used to speed up the surface methods.

The order of B-splines along the surface parameters coincides with the order of the divided difference used to compute the corresponding B-splines. The knot vectors *uknots* and *vknots* are nondecreasing sequences of real numbers and define the domain of the surface parameters and its shape. Let the knot vector *uknots* contain *uknotsCount* elements, and let the number of points in each row of the rectangular grid be equal to *ucount*. For a NURBS surface that is nonperiodic in the first parameter, the numbers of elements in the sets are related by the equality *ucount* + *degree* = *uknotsCount*. For a periodic NURBS surface, the numbers of elements in the sets are related by the equality *knotsCount* + 2 *degree* − 1 = *knotsCount*.

In the :blue:`PointOn`\( double *u*, double *v*, :ref:`MbCartPoint3D <MbCartPoint3D_Radius_Vector_of_Point_in_3D_Space_>` & **s** ) method, the radius vector of the surface **s** is described by the vector function

.. math::


   r(u, v) = \frac{\sum\limits_{i = 0}^{\mathit{\text{vcount}} - 1}{\sum\limits_{j = 0}^{\mathit{\text{ucount}} - 1}{N_i^{vdegree}(v) N_j^{udegree}(u) \mathit{\text{weight}}\lbrack i\rbrack\lbrack j\rbrack \mathbf{\text{points}}\lbrack i\rbrack\lbrack j\rbrack}}}{\sum\limits_{i = 0}^{\mathit{\text{vcount}} - 1}{\sum\limits_{j = 0}^{\mathit{\text{ucount}} - 1}{N_i^{vdegree}(v) N_j^{udegree}(u) \mathit{\text{weight}}\lbrack i\rbrack\lbrack j\rbrack}}},


where :math:`N_i^{vdegree}(v)` and :math:`N_j^{udegree}(u)` are B-splines.

Fig. 1 shows segments connecting neighboring control points **points**\[*i*]\[*j*].

.. figure:: /MODELER/SURFACES/pictures/O.5.22.1.png
   :width: 12cm
   :align: center

   Fig. 1


Fig. 1 illustrates the conventional arrangement of the control points at the nodes of a rectangular grid. The NURBS surface constructed on the same control points is shown in Fig. 2:

.. figure:: /MODELER/SURFACES/pictures/O.5.22.2.png
   :width: 12cm
   :align: center

   Fig. 2


The curves on the surface **s**\(*const*, *v*) and **s**\(*u*, *const*) with parameters *u* = const or *v* = const are B-curves of degree *udegree* and *vdegree* respectively. In the first parameter, the surface has order *udegree*, and in the second parameter, the surface has order *vdegree*. The domain of the parameters of a nonperiodic NURBS surface is a rectangle: *uknots*\[*udegree* − 1] ≤ *u* ≤ *uknots*\[*ucount*], *vknots*\[*vdegree* − 1] ≤ *v* ≤ *vknots*\[*vcount*].

The surface can be periodic both in the first parameter and in the second parameter. For a periodically closed surface, the knot vectors *uknots* and *vknots* have *udegree* − 1 and *vdegree* − 1 additional elements respectively. The domain of the parameters of a NURBS surface that is periodic in both parameters is a rectangle: *uknots*\[*udegree* − 1] ≤ *u* ≤ *uknots*\[*ucount* + *udegree* − 1], *vknots*\[*vdegree* − 1] ≤ *v* ≤ *vknots*\[*vcount* + *vdegree* − 1].

Each surface can construct its NURBS copy by the virtual method :blue:`NurbsSurface`\( const MbNurbsParameters & *uParam*, const MbNurbsParameters & *vParam* ).