====== Curves ======

===== Introduction =====

    * ''Curves'' are defined using ''[[doc:user:geometry:user:points]]'' and auxiliary data, and are used to define ''[[doc:user:geometry:user:contours]]''. 
    * A curve orientation is given by the succession if its points. For example, a straight line is oriented from its first point towards its second point. This orientation is relevant when defining contact matrices, for instance. For 2D rigid-deformable contact, a curve must be defined with "area to the left", which means that the normal must point towards the inside of the matter (also applicable with ''Surfaces'' in 3D).
    * The normal of a curve is defined such as $\boldsymbol{t} \wedge \boldsymbol{n} = (0,0,1)$.

{{ doc:user:doc-courbe.png?300 |}}

===== Line: Straight Segment =====

{{ doc:user:doc-droite.png |}}

A ''Line'' (line segment) is defined with its two vertices.

  line = curveset.add( Line(number, pt1, pt2) )	

with
| ''number''          | user number (unique among Curves and $\ge 1$)  |
| ''pt1'',''pt2''     | the 2 ''Points'' used as vertices              |

===== Arc: Arc of Circle ===== 

{{ doc:user:doc-arc.png |}}

An ''Arc'' of circle is defined using three points, as shown on the figure.

  arc = curveset.add( Arc(number, pt1, pt2, pt3) )

with
| ''number''                  | user number (unique among Curves and $\ge 1$)  |
| ''pt1'',''pt2'',''pt3''     | the 3 ''Points''              |


===== Cubic Spline =====

{{ doc:user:doc-spline.png |}}

==== "Open" Cubic Spline ====

  spl = curveset.add( CubicSpline(number, [pt1, pt2, pt3, pt4]) )	
  spl.useLsTangent()    # tangents are computed 
                        # using local reconstruction [DEFAULT]
  spl.useLittTangents() # tangents are computed 
                        # using Litt/Beckers lectures
                      
with
| ''number''          | user number (unique among Curves and $\ge 1$)  |
| ''pt1'',''pt2'',''pt3'',...     | the list of ''Points''              |

==== "Closed" Cubic Spline ====

To close a spline, the first and last points of the list must be the same.

  spl = CubicSpline(number, [pt1, pt2, pt3, pt4, ..., pt1])

==== Spline-reconstruction based on a mesh ====

{{:doc:user:ico-advanced.png?40 |Advanced}}
It is possible to construct a spline based on the mesh of a line. This way, a smooth approximation of this mesh if obtained.

  spl = CubicSpline(number, obj)

where ''obj'' is a meshed object.

===== Circle: Full Circle ===== 

{{ doc:user:doc-cercle.png |}}

A circle is defined with its center and radius (this function is only defined in the $z=0$ plane)

  circ2d = curset.add( Circle(number, centre, R) )

with
| ''number''  | user number (unique among Curves and $\ge 1$)  |
| ''centre''  | centre ''Point''                               |
| ''R''       | radius                                         |

The orientation of the ''Circle'' can be inverted (and so will its tangent and normal used for contact):
  circ2d.reverse()      



===== NURBS =====

A Non-Uniform Rational Basis Spline (N.U.R.B.S.) is defined as:

{{ doc:user:doc-nurbs.png |}}

  nur = curset.add( NurbsCurve(number) )
  nur.setDegree(degree) 	 
  nur.push(pt1); nur.pushWeight(weight1)
  nur.push(pt2); nur.pushWeight(weight2) 	 
  nur.pushKnot(knot1) 	 
  nur.pushKnot(knot2) 	 

where
|< 30em - >|
| ''number'' | user number (unique among Curves and $\ge 1$)  |
| ''pt1'', ''pt2'' | ''[[doc:user:geometry:user:points]]'' used as supports | 
| ''degree'' | degré de la coube | 
| ''weight1'', ''weight2'' | weights| 
| ''knot1'', ''knot2'' | knot vector | 


===== PythonCurve: Python-interpreted Curves =====
{{:doc:user:ico-advanced.png?40 |Advanced}}

If a ''Curve'' is not implemented in Metafor, it can be programmed in [[doc:user:general:glossaire#python]] using the generic Curve called ''PythonCurve''.

The method ''.push()'' is used to add points.

''PythonCurve'' possesses four member functions that can be overloaded. The method named ''setEval(fct)'' is used to defined the evaluation [[doc:user:tutorials:tuto0#fonctions|python function]]. The methods named ''setTg'', ''setDTg'' and''setLen()'' respectively define the tangent, its derivative and the curvilinear abscissa. The function ''setEval(fct)'' is the only one required to mesh the curve, when the other three are used for contact.

__Example:__ 

See ''toolbox.curves'' which defines a ''Parabola''. The input file ''apps.qs.parabola'' is an example of application.

  from toolbox.curves import Parabola
  parab = curset.add(Parabola(1, pt1, pt6, pt2))

These lines create a parabola #1, based on the three previously defined points ''pt1'', ''pt6'' and ''pt2''.