.. role:: blue
   :class: blue

MbCurveSpiral Variable Radius Spiral
====================================
.. _MbCurveSpiral_Variable_Radius_Spiral_:

.. kernel version: 

The :blue:`MbCurveSpiral` class is declared in the *cur_curve_spiral.h* file.

The variable radius spiral is derived from the MbSpiral curve.
The spiral is defined by the local coordinate system
:ref:`MbPlacement3D <MbPlacement3D_Local_Coordinate_System_>` **position**, the two-dimensional curve
**curve** specifying the radius law, the spiral pitch
*step*, 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 two-dimensional curve **curve**
lies in the XZ plane of the local coordinate system and defines
the radius variation law of the spiral. The **position.axisZ** axis serves as the
abscissa axis, and the **position.axisX** axis serves as the ordinate axis of the
two-dimensional space of the **curve**. The origin of the two-dimensional coordinate
system of the **curve** coincides with the origin of the local
coordinate system **position**. The spiral radius is equal to the ordinates of points
of the two-dimensional curve **curve**. 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.

In the :blue:`PointOn`\( double & *t*,
:ref:`MbCartPoint <MbCartPoint_Point_Radius_Vector_in_Two-Dimensional_Space_>` & **r**) method, the radius vector of the curve **r** is described by the vector function

.. math::

   \mathbf{r}(t) = \mathbf{position.origin} +
   radius(t) \left( \cos(t) \cdot \mathbf{position.axisX} +
   \sin(t) \cdot \mathbf{position.axisY} \right) +
   \left( t \cdot \frac{step}{2\pi} \right) \mathbf{position.axisZ},

where *radius*\(\ *t*) is the local radius. The local radius
*radius*\(\ *t*) is computed as follows. For a given
parameter *t*, the abscissa
*t*·*step* / 2π of the required two-dimensional point
of the curve is determined. Then the intersection point of the **curve**
and the vertical line intersecting the abscissa axis at
*t*·*step* / 2π is found. The ordinate of the
two-dimensional intersection point of **curve** and the vertical line is equal to
the required spiral radius *radius*\(\ *t*). The local radius is equal to
the distance from the local abscissa axis to the intersection point of the vertical
line with the **curve** in the two-dimensional space of the XZ
plane of the local coordinate system **position**.

The variable radius spiral is shown in the figure:

.. figure:: /MODELER/CURVES/pictures/O.4.15.1.png
   :width: 9cm
   :align: center

The **curve** must lie above the abscissa
axis and must not intersect the abscissa axis of its two-dimensional coordinate
system. The **curve** must not have vertical
tangents. 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.