.. role:: blue
   :class: blue

MbConeSurface Conical Surface
==============================
.. _MbConeSurface_Conical_Surface_:

.. версия ядра: 

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

The :blue:`MbConeSurface` conical surface belongs to the group of elementary
surfaces :blue:`MbElementarySurface`. The conical surface is described by
the radius *radius*, the height *height*, and the cone angle
*angle*, 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 straight line 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 the origin of the local coordinate system, and the value
*v* = 1 corresponds to the point at a distance *height* from the XY plane
of the local coordinate system of the surface.

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{radius} + \mathit{height} \cdot v \cdot \tan(\mathit{angle}) \right)
   \left( \cos(u)\, \mathbf{position.axisX} + \sin(u)\, \mathbf{position.axisY} \right) +
   \mathit{height} \cdot v\, \mathbf{position.axisZ}
   \end{equation}

Fig. 1 shows the conical surface:

.. figure:: /MODELER/SURFACES/pictures/O.5.4.1.png
   :width: 15cm
   :align: center

   Fig. 1

The radius and height must be greater than zero, and the absolute value of the angle must not
exceed π/2: *radius* > 0, *height* > 0,
-π/2 < *angle* < π/2. When *angle* = 0, the conical
surface is equivalent to a cylindrical surface. The following inequalities must hold for the boundary
parameters of the surface: *umin* < *umax*, *vmin* < *vmax*.
The pole of the surface corresponds to the value of the second parameter
*v* = -*radius* / (*height* · tg(*angle*)). The boundary
parameters *vmax* and *vmin* take values such that
the surface is located on one side of the pole.

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
towards the convex side of the surface (away from the surface axis); if the local
coordinate system is left-handed, the normal is directed towards the concave side
of the surface (towards the surface axis).