.. role:: blue
   :class: blue

MbTorusSurface Toroidal Surface
================================
.. _MbTorusSurface_Toroidal_Surface_:

.. версия ядра: 1180ХХ

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

The :blue:`MbTorusSurface` toroidal surface belongs to the group of elementary
surfaces :blue:`MbElementarySurface`. The toroidal surface is described by the major
radius *majorRadius* and the tube radius *minorRadius*,
defined in the local coordinate system
:ref:`MbPlacement3D <MbPlacement3D_Local_Coordinate_System_>` **position**.

The first surface parameter is measured along the arc from the vector
**position.axisX** in the direction of the vector **position.axisY**.
The first surface parameter *u* takes values on the interval:
*umin* ≤ *u* ≤ *umax*. The values *u* = 0 and *u* = 2π
correspond to a point on the XZ plane. The surface can be
periodic with respect to the first parameter. For a periodic surface,
*umax* - *umin* = 2π; for a non-periodic surface,
*umax* - *umin* < 2π.

The second surface parameter is measured along the arc from the XY plane
of the local coordinate system of the surface in the direction of the vector
**position.axisZ**. The second surface parameter *v* takes
values on the interval: *vmin* ≤ *v* ≤ *vmax*. The values
*v* = 0 and *v* = 2π correspond to a point on the XY plane of the local
coordinate system of the surface. The surface can be periodic with respect to
the second parameter when
*majorRadius* > *minorRadius*. For a periodic surface,
*vmax* - *vmin* = 2π; for a non-periodic surface,
*vmax* - *vmin* < 2π.

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

.. math::
   :nowrap:

   \begin{equation}
   \mathbf{s}(u, v) = \mathbf{position.origin} +
   \left( \mathit{majorRadius} + \mathit{minorRadius} \cdot \cos(v) \right)
   \left( \cos(u)\, \mathbf{position.axisX} + \sin(u)\, \mathbf{position.axisY} \right) + \\
   \mathit{minorRadius} \cdot \sin(v)\, \mathbf{position.axisZ}
   \end{equation}

Figure shows the toroidal surface:

.. figure:: /MODELER/SURFACES/pictures/O.5.6.1.png
   :width: 15cm
   :align: center

The tube radius must be greater than zero: *minorRadius* > 0.
The major radius must be no less than the tube radius taken with the opposite
sign: *majorRadius* > -*minorRadius*. If
*majorRadius* < *minorRadius*, the surface has
a pole at the parameter values
*v* = arccos(*majorRadius* / *minorRadius*) and
*v* = 2π - arccos(*majorRadius* / *minorRadius*). The following inequalities must hold
for the boundary parameters of the surface:
*umin* < *umax*, *vmin* < *vmax*.

The local coordinate system **position** can be either right-handed or
left-handed. If the local coordinate system is right-handed, the normal is directed
away from the tube of the surface; if the local coordinate system is left-handed,
the normal is directed inward into the tube of the surface.