.. role:: blue
   :class: blue

MbContourOnSurface Contour on Surface
=====================================
.. _MbContourOnSurface_Contour_on_Surface_:

.. kernel version:

The :blue:`MbContourOnSurface` class is declared in the *cur_contour_on_surface.h* file.

A contour on surface MbContourOnSurface is defined by a surface
:ref:`MbSurface <MbSurface_Surface_>` **surface** and a two-dimensional contour
:ref:`MbContour <MbContour_Two-Dimensional_Contour_>` **contour** in the surface parameter
space. The curve also contains some additional data that is not mandatory
and is used to speed up the work of the curve methods.

A contour on a surface is a composite curve, therefore it may
have breaks at the junction points of the segments of the two-dimensional contour. A contour on
the surface represents a mapping of a contour from the two-dimensional
surface parameter space into three-dimensional space. The two-dimensional
contour **contour** may lie outside the domain
of the surface parameters. The parameter domain
of the contour on the surface coincides with the parameter domain
of the two-dimensional contour **contour**.

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{surface}(\mathbf{point}.x, \mathbf{point}.y),

where *point* is a point of the two-dimensional contour that is
calculated by the *contour* → :blue:`PointOn`\( *t*, *point* ) method.
The *x* and *y* coordinates of the two-dimensional point *point*
serve as the *u* and *v* parameters of the surface
*surface*\( *u*, *v* ). A contour on the surface is constructed
by introducing a dependence of the surface parameters *u* and *v* on
a common parameter *t*:
*u* = *u*\( *t* ), *v* = *v*\( *t* ). This
dependence is defined by the two-dimensional contour *contour*.
The derivative of a contour on a surface is computed as a composite function

.. math::

   \frac{\mathit{dr}(t)}{\mathit{\text{dt}}} = \frac{\partial\mathbf{\text{surface}}(u,v)}{\partial u}\mathbf{\text{derive}}\text{.}{x + \frac{\partial\mathbf{\text{surface}}(u,v)}{\partial v}}\mathbf{\text{derive}}\text{.}y,

where **derive** is the derivative of the two-dimensional contour that is
calculated by the *contour* → :blue:`FirstDer`\( *t*, *derive* ) method.
The derivative of the contour on the surface lies in the tangent plane
of the surface constructed at the point under consideration.

A contour on the surface is shown in the figure:

.. figure:: /MODELER/CURVES/pictures/O.4.22.1.png
   :width: 12cm
   :align: center

A contour on a surface can be periodic if the two-dimensional contour
*contour* is periodic or if the surface **surface** is periodic and
the end points of the contour **contour** are shifted by the corresponding period
of the surface that is periodic in the first or second parameter
**surface**.

A periodic contour on a surface is usually used to describe
the boundary of this surface.

The surface *surface* of a contour on a surface can be any
surface except for a surface bounded by curves
:ref:`MbCurveBoundedSurface <MbCurveBoundedSurface_Surface_with_Arbitrary_Borders_>`. If a contour on a surface
bounded by curves needs to be constructed, then the base surface
*MbCurveBoundedSurface* is used as the surface.