.. role:: blue
   :class: blue

MbExpansionSurface Motion Surface
==================================
.. _MbExpansionSurface_Motion_Surface_:

.. версия ядра: 1180ХХ

The :blue:`MbExpansionSurface` class is declared in the *surf\_expansion\_surface.h* file.

The :blue:`MbExpansionSurface` motion surface belongs to the group of
swept surfaces :blue:`MbSweptSurface`. The motion surface is a special case
of a swept surface with a curvilinear spine curve. The motion surface is
described by the generatrix curve
:ref:`MbCurve3D <MbCurve3D_Curve_>` * **curve**, the spine curve
:ref:`MbCurve3D <MbCurve3D_Curve_>` * **spine**, and the point at the start of the spine
:blue:`MbCartPoint3D` **origin**. The surface is formed by moving the
generatrix curve along the spine curve. In a special case, the generatrix
curve of the motion surface can change its shape. In the latter case, the
curve data contain the second generatrix curve
**brink**, the point at the end of the spine **ending**, the initial
parameter *tmin* of the **brink** curve, and the derivative *dt*
of the parameter of the **brink** curve with respect to the parameter of the
generatrix curve **curve**. The derivative *dt* is defined by

.. math::
   :nowrap:

   \begin{equation}
   \mathit{dt} = \frac{\mathit{tmax} - \mathit{tmin}}{\mathit{umax} - \mathit{umin}},
   \end{equation}

where *tmax* is the terminal parameter of the **brink** curve. In the general
case, the pointer to the second generatrix curve **brink** can be null, which
means that the second generatrix curve is absent.

The first surface parameter *u* coincides with the parameter of the
generatrix curve **curve**. The first surface parameter takes values on
the interval *umin* ≤ *u* ≤ *umax*, which corresponds to the domain
of the generatrix curve. The surface can be periodic with respect to
the first parameter if the generatrix curve is periodic.

The second surface parameter *v* coincides with the parameter of the spine
curve and takes values on the interval
*vmin* ≤ *v* ≤ *vmax*. The surface can be periodic
with respect to the second parameter if the spine curve is periodic and
the second generatrix curve is absent.

In the :blue:`PointOn`\( double *u*, double *v*,
:blue:`MbCartPoint3D` & **s** ) method, in the general case, the position vector
of the surface **s** is described by the vector function

.. math::
   :nowrap:

   \begin{equation}
   \mathbf{s}(u, v) = \mathbf{spine}(v) + \mathbf{curve}(u) - \mathbf{origin}
   \end{equation}

The motion surface in the general case is shown in Fig. 1.

.. figure:: /MODELER/SURFACES/pictures/O.5.9.1.png
   :width: 12cm
   :align: center

   Fig. 1

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

.. math::
   :nowrap:

   \begin{equation}
   \mathbf{s}(u, v) = \mathbf{spine}(v) +
   \left( \mathbf{curve}(u) - \mathbf{origin} \right) (1 - w) +
   \left( \mathbf{brink}(t) - \mathbf{ending} \right) w,
   \end{equation}

where

.. math::
   :nowrap:

   \begin{equation}
   w = \frac{v - \mathit{vmin}}{\mathit{vmax} - \mathit{vmin}}, \quad
   t = \mathit{tmin} + (u - \mathit{umin}) \cdot \mathit{dt}.
   \end{equation}

The motion surface in the special case is shown in Fig. 2:

.. figure:: /MODELER/SURFACES/pictures/O.5.9.1.png
   :width: 12cm
   :align: center

   Fig. 2

To avoid self-intersections of the surface, the generatrix and spine
curves must not have segments that are parallel to each other. In certain
cases, the motion surface may have singular points.