.. role:: code
   :class: code

Two-Dimensional Projection Curve MbProjCurve
===========================================
.. _MbProjCurve_Two-Dimensional_Projection_Curve_:

.. kernel version: 118065 

The :code:`MbProjCurve` class is declared in the *cur_projection_curve.h* file.

A two-dimensional projection curve is described by a space curve
:ref:`MbCurve3D <MbCurve3D_Curve_>`\* **spaceCurve**, a surface
:ref:`MbSurface <MbSurface_Surface_>`\* **surface** and a two-dimensional curve
:ref:`MbCurve <MbCurve_Two-Dimensional_Curve_>`\* **curve**. The curve also has some additional data that is optional and is used to speed up the curve methods.

A two-dimensional projection curve represents the projection of the
space curve **spaceCurve** onto the surface
**surface**, which is approximately described by the two-dimensional curve
**curve** within the definition domain of the surface parameters.
The definition domains of the parameters of the curves **spaceCurve** and
**curve** coincide. The two-dimensional curve **curve**
is usually a spline whose control points are obtained by projecting points
of the space curve **spaceCurve** onto the
surface **surface**. The parametrization of the curve **curve** is consistent with the parametrization of the space curve at
the control points. The two-dimensional curve **curve** may
lie outside the definition domain of the surface parameters.

In the method

.. code:: cpp
  
   PointOn( double & t, MbCartPoint & r) 

the radius vector of the curve **r** is described by the vector function

.. math::

   \mathbf{r}(t) = [u \quad v],

where *u*, *v* are the parameters of the projection of the point
**spaceCurve**\(*t*) onto the surface **surface**. 

The :code:`PointOn`\() method corrects the parameter *t* if it is outside
the curve domain. The :code:`_PointOn`\() method is used to calculate a point
without correcting the parameter.

An initial approximation of the parameters *u* and *v* is computed
by the method 

.. code:: cpp

   curve->PointOn(t, point), 

*u* = **point**\ .x, *v* = **point**\ .y.


Then the parameters *u* and *v* are refined by an iterative method
using the equations

.. math::

   \mathbf{deriveU} \cdot (\mathbf{spaceCurve}(t) - \mathbf{surface}(u,\, v)) = 0,

   \mathbf{deriveV} \cdot (\mathbf{spaceCurve}(t) - \mathbf{surface}(u,\, v)) = 0,

where **deriveU** and **deriveV** are the partial derivatives
of the radius vector of the surface, which are calculated by the methods

.. code:: cpp

   surface->_DeriveU(u, v, deriveU)

and

.. code:: cpp

   surface->_DeriveV(u, v, deriveV),

respectively.

A projection curve is shown in the figure:

.. figure:: ../../pictures/O.3.16.1.png
   :width: 15cm
   :align: center

The projection curve is used for an exact description of the projection
of a space curve onto a surface.