.. role:: blue
   :class: blue

MbSphereSurface Spherical Surface
===================================
.. _MbSphereSurface_Spherical_Surface_:

.. версия ядра: 1180ХХ

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

The :blue:`MbSphereSurface` sphere belongs to the group of elementary surfaces
:blue:`MbElementarySurface`. The sphere is described by the radius *radius*, 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 value
*v* = 0 corresponds to a point on the XY plane of the local coordinate
system of the surface. The surface is non-periodic with respect to the second parameter.

In the :blue:`PointOn`\( double *u*, double *v*,
:blue:`MbCartPoint3D` & **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} +
   \mathit{radius} \left( \cos(u)\, \mathbf{position.axisX} + \sin(u)\, \mathbf{position.axisY} \right) +
   \mathit{radius} \cdot \sin(v)\, \mathbf{position.axisZ}
   \end{equation}

Fig. 1 shows the sphere:

.. figure:: /MODELER/SURFACES/pictures/O.5.5.1.png
   :width: 15cm
   :align: center

   Fig. 1

The sphere radius must be greater than zero: *radius* > 0. The sphere
has poles at parameter values *v* = π/2 and *v* = -π/2. The following inequalities must hold
for the boundary parameters of the surface:
*umin* < *umax*, *vmin* < *vmax*,
*vmax* ≤ π/2, *vmin* ≥ -π/2.

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
outward from the sphere; if the local coordinate system is left-handed, the normal
is directed inward into the sphere.