.. role:: blue
   :class: blue

MbConeSpiral Conical Spiral
===========================
.. _MbConeSpiral_Conical_Spiral_:

.. kernel version: 

The :blue:`MbConeSpiral` class is declared in the *cur_cone_spiral.h* file.

The conical spiral is derived from the :blue:`MbSpiral` curve. The spiral
is defined by the local coordinate system
:ref:`MbPlacement3D <MbPlacement3D_Local_Coordinate_System_>` **position**, the radius *radius*, the tangent
of the cone angle *tgAlpha*, the spiral pitch divided by 2π,
*step_2pi*, and two boundary parameters of the spiral *tmin* and
*tmax*. The curve also has some additional data that is optional
and is used to speed up the curve methods.

The spiral axis coincides with the **position.axisZ**
axis of the local coordinate system. The *tgAlpha* parameter is equal to the tangent
of the angle between the spiral axis and the cone generatrix of the spiral. The *tmin* and
*tmax* parameters are angles measured from the
**position.axisX** vector to the **position.axisY** vector. Angle
values that are multiples of 2π correspond to points of the curve in the XZ plane of the local
coordinate system. The curve parameter *t* takes values on the
segment *tmin* ≤ *t* ≤ *tmax*.

In the :blue:`PointOn`\( double & *t*,
:blue:`MbCartPoint3D` & **r**) method, the radius vector of the curve **r** is described by the vector function

.. math::

   \mathbf{r}(t) = \mathbf{position.origin} +
   \left( radius + t \cdot tgAlpha \cdot step\_2pi \right)
   \left( \cos(t) \cdot \mathbf{position.axisX} + \sin(t) \cdot \mathbf{position.axisY} \right) +
   \left( t \cdot step\_2pi \right) \mathbf{position.axisZ}.

The conical spiral is shown in the figure:

.. figure:: /MODELER/CURVES/pictures/O.4.14.1.png
   :width: 9cm
   :align: center

The curve radius must be greater than zero: *radius* > 0. For
the boundary parameters, the inequality
*tmin* < *tmax* must hold. The curve cannot be periodic.

The local coordinate system **position** can be either right-handed or
left-handed. For a cylindrical spiral, *tgAlpha* = 0.