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ULiege - Aerospace & Mechanical Engineering

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# Curves

## Introduction

• Curves are defined using Points and auxiliary data, and are used to define Wires.
• 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 $\boldsymbol{t} \wedge \boldsymbol{n} = (0,0,1)$.

## Line: straight segment

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

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

with

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

## Arc

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

### "Open" cubic spline

spl = curveset.add( CubicSpline(number, [pt1, pt2, pt3, pt4]) )
spl.useLsTangent()    # tangents are calculated
# using local reconstruction [DEFAULT]
spl.useLittTangents() # tangents are calculated
# using Litt/Beckers lectures


### "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 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.

## Full Circle

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

circ2d = curset.add( Circle(number, pt1, 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:

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

 number curve number pt1, pt2 Points used as supports degree degré de la coube weight1, weight2 weights knot1, knot2 knot vector closed Boolean to determine whether the Nurb is closed obj GObject support of a topology (curve,wire,group)

## PythonCurve: Python-interpreted Curves

If a curve cannot be defined with the functions above, it can be programmed in python using the generic Curve called PythonCurve.

The method .push() is used to add points, and it possesses four member functions that can be overloaded. The method named setEval(fct) is used to defined the evaluation python function. The methods named setTg, setDTg andsetLen() 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. 