.. role:: code
   :class: code


MbCurve3D Curve
===============
.. _MbCurve3D_Curve_:

.. kernel version: 118065

The :code:`MbCurve3D` class is declared in the *curve3d.h* file.

The :code:`MbCurve3D` curve is derived from the
:ref:`MbSpaceItem <MbSpaceItem_Three-Dimensional_Geometrical_Object_>` and
:code:`MbNestSyncItem` classes:


.. figure:: ../../pictures/O.4.1.1.png
   :width: 15cm
   :align: center


A curve is an abstract class. The C3D geometric kernel implements the
following curves derived from the :code:`MbCurve3D` class:


:ref:`MbLine3D <MbLine3D_Straight_Line_>` -- a straight line;

:ref:`MbLineSegment3D <MbLineSegment3D_Straight_Line_Segment_>` -- a line segment;

:ref:`MbArc3D <MbArc3D_Elliptical_Arc_>` -- an elliptical arc;

:ref:`MbPolyline3D <MbPolyline3D_Polyline_>` -- a polyline;

:ref:`MbNurbs3D <MbNurbs3D_NURВS-Curve_>` -- a B-curve (NonUniform Rational B-Spline);

:ref:`MbBezier3D <MbBezier3D_Bezier_Composite_Curve_>` -- a composite
Bezier curve;

:ref:`MbHermit3D <MbHermit3D_Hermite_Curve_>` -- a Hermite curve;

:ref:`MbCubicSpline3D <MbCubicSpline3D_Cubic_Spline_>` -- a cubic
spline;

:ref:`MbOffsetCurve3D <MbOffsetCurve3D_Equidistant_Curve_>` -- an offset
curve;

:ref:`MbTrimmedCurve3D <MbTrimmedCurve3D_Trimmed_Curve_>` -- a trimmed
curve;

:ref:`MbReparamCurve3D <MbReparamCurve3D_Reparametrized_Curve_>` -- a
reparameterized curve;

:ref:`MbConeSpiral <MbConeSpiral_Conical_Spiral_>` -- a conical
spiral;

:ref:`MbCurveSpiral <MbCurveSpiral_Variable_Radius_Spiral_>` -- a spiral
with a straight axis and a variable radius;

:ref:`MbCrookedSpiral <MbCrookedSpiral_Spiral_with_Curved_Planar_Axis_>` -- a
spiral with an axis in the form of a planar curve;

:ref:`MbBridgeCurve3D <MbBridgeCurve3D_Joining_Curve_>` -- a Hermite
spline joining two curves;

:ref:`MbContour3D <MbContour3D_Contour_>` -- a contour
(a composite curve);

:ref:`MbPlaneCurve <MbPlaneCurve_Plane_Curve_>` -- a planar curve in space;

:ref:`MbSurfaceCurve <MbSurfaceCurve_Curve_on_Surface_>` -- a curve on a surface;

:ref:`MbSilhouetteCurve <MbSilhouetteCurve_Silhouette_Curve_>` -- a surface
silhouette curve;

:ref:`MbContourOnSurface <MbContourOnSurface_Contour_on_Surface_>` -- a contour on a surface;

:ref:`MbContourOnPlane <MbContourOnPlane_Contour_on_Plane_>` -- a contour on a plane;

:ref:`MbSurfaceIntersectionCurve <MbSurfaceIntersectionCurve_Surface_Intersection_Curve_>` --
a surface intersection curve.

The :code:`MbCurve3D` curve is a vector function


.. math::

   \mathbf{\text{curve}}(t) =
   \begin{bmatrix}
      x(t) & y(t) & z(t)
   \end{bmatrix}


of a scalar parameter *t* taking values within the interval
[*t*\ :subscript:`min`, *t*\ :subscript:`max`]. A curve is a continuous
mapping of a part of the numerical axis into three-dimensional space.
The range of the curve parameter is the interval
[*t*\ :subscript:`min`, *t*\ :subscript:`max`] in one-dimensional space.
The coordinates *x*\(*t*\), *y*\(*t*\), *z*\(*t*\) of the curve point
**curve**\(*t*\) are single-valued continuous functions of the parameter
*t*.

The boundary values *t*\ :subscript:`min` and
*t*\ :subscript:`max` of the parameter domain are returned by the
curve methods double :code:`GetTMin`\() and double :code:`GetTMax`\(),
respectively.


A curve is called periodic if there exists
*p* > 0 such that
**curve**\(*t* ± *kp*) = **curve**\(*t*\), where
*k* is an integer.


The methods


.. code-block:: cpp

   bool IsPeriodic()


and


.. code-block:: cpp

   bool IsClosed()


of a periodic curve return *true*.


The method


.. code-block:: cpp

   double GetPeriod()


of a periodic curve or a curve that can be extended to a periodic one
returns the period *p*. The parameter domain of a periodic curve always
lies within a single period.

A curve is considered periodic if:

- The start and end points of the curve coincide.
  The following 3D curve types are not considered periodic when their start
  and end points coincide, for example, in the case of a degenerate curve;
  the :code:`IsClosed`\() and :code:`IsPeriodic`\() methods always return
  *false* for them:


  - :ref:`MbLine3D <MbLine3D_Straight_Line_>`;
  - :ref:`MbLineSegment3D <MbLineSegment3D_Straight_Line_Segment_>`;
  - :code:`MbSpiral` and derived classes;
  - :ref:`MbBridgeCurve3D <MbBridgeCurve3D_Joining_Curve_>`;
  - :ref:`MbContour3D <MbContour3D_Contour_>`, if it contains only one segment
    of one of the types listed above.


- The derivatives at the start and end points coincide.
  If a curve can inherently have breaks (a contour or a polyline), the
  derivatives are allowed not to coincide.
  For a Bezier curve, the derivatives must coincide in direction but may
  differ in magnitude.


To check whether the start and end points of a curve coincide with a specified
tolerance, use the :code:`IsTouch`\() method, which determines coincidence
regardless of the smoothness of the closure.


The primary curve method is


.. code-block:: cpp

   void PointOn( double &        t,
                 MbCartPoint3D & r )


It returns the radius-vector **r** of the curve point for a given parameter
*t*.


The methods


.. code-block:: cpp

   void FirstDer( double &      t,
                  MbVector3D & rt )


.. code-block:: cpp

   void SecondDer( double &       t,
                   MbVector3D & rtt )


.. code-block:: cpp

   void ThirdDer( double &        t,
                  MbVector3D & rttt )


return the first **r**\ :subscript:`t`, second
**r**\ :subscript:`tt`, and third
**r**\ :subscript:`ttt` derivatives, respectively, of the curve radius-vector
for a given parameter *t*. These methods correct the curve parameter when it
is outside the parameter domain (except for the :code:`MbLine3D` line). When
the curve parameter *t* is outside the interval
[*t*\ :subscript:`min`, *t*\ :subscript:`max`], non-periodic curves shift
the parameter *t* to the nearest boundary
*t*\ :subscript:`min` or *t*\ :subscript:`max`, while periodic curves add
or subtract the required number of periods.


The method


.. code-block:: cpp

   void _PointOn( double          t,
                  MbCartPoint3D & r )


returns the radius-vector **r** of the curve point for a given parameter
*t* both within and outside the curve parameter domain. In the general case,
a non-periodic curve outside its parameter domain is extended along the tangent
at the boundary point. The exceptions are periodic curves, the
:code:`MbArc3D` arc, :code:`MbSpiral` spirals, the :code:`MbCharacterCurve3D`
symbolic curve, and the :code:`MbTrimmedCurve3D` trimmed curve within its base
curve. Periodic curves outside their parameter domain are extended cyclically.
The methods


.. code-block:: cpp

   void _FirstDer( double &      t,
                   MbVector3D & rt )


.. code-block:: cpp

   void _SecondDer( double &       t,
                    MbVector3D & rtt )


.. code-block:: cpp

   void _ThirdDer( double &        t,
                   MbVector3D & rttt )


return the first **r**\ :subscript:`t`, second
**r**\ :subscript:`tt`, and third
**r**\ :subscript:`ttt` derivatives, respectively, of the curve radius-vector
for a given parameter *t* both within and outside the curve domain.


Curves override the following methods of a three-dimensional geometric object:


methods for transforming a geometric object:


.. code-block:: cpp

   void Move( const MbVector3D &      to,
                    MbRegTransform *  iReg = nullptr )


.. code-block:: cpp

   void Rotate( const MbAxis3D &       axis,
                      double           angle,
                      MbRegTransform * iReg = nullptr )


.. code-block:: cpp

   void Transform( const MbMatrix3D &     matr,
                        MbRegTransform *  iReg = nullptr )


methods for copying objects, checking equality and similarity, and making
objects equal:


.. code-block:: cpp

   MbSpaceItem & Duplicate( MbRegDuplicate * iReg = nullptr )


.. code-block:: cpp

   bool IsSame( const MbSpaceItem & other,
                      double        accuracy = LENGTH_EPSILON )


.. code-block:: cpp

   bool IsSimilar( const MbSpaceItem & item )


.. code-block:: cpp

   bool SetEqual( const MbSpaceItem & item )


methods that return a type from the geometric object enumeration:


.. code-block:: cpp

   MbeSpaceType IsA()


.. code-block:: cpp

   MbeSpaceType Type()


.. code-block:: cpp

   MbeSpaceType Family()


methods for retrieving and editing the internal object data:


.. code-block:: cpp

   MbProperty & CreateProperty( MbePrompt name )


.. code-block:: cpp

   void GetProperties( MbProperties & properties )


.. code-block:: cpp

   void SetProperties( MbProperties & properties )


the method that builds a polygonal copy of a geometric object:


.. code-block:: cpp

   void CalculateMesh( const MbStepData & stepData,
                       const MbFormNote & note,
                             MbMesh &     mesh ) const;


All curves, except for
:ref:`MbContour3D <MbContour3D_Contour_>`,
:ref:`MbContourOnSurface <MbContourOnSurface_Contour_on_Surface_>`,
and :ref:`MbContourOnPlane <MbContourOnPlane_Contour_on_Plane_>`,
usually do not have breaks.
:ref:`MbContour3D <MbContour3D_Contour_>`,
:ref:`MbContourOnSurface <MbContourOnSurface_Contour_on_Surface_>`,
and :ref:`MbContourOnPlane <MbContourOnPlane_Contour_on_Plane_>`
are composite curves and can have breaks at the junctions of their
constituent segments.