gen_contour_nurbs_xld (Operator)

Name

gen_contour_nurbs_xld — Transform a NURBS curve into an XLD contour.

Signature

gen_contour_nurbs_xld( : Contour : Rows, Cols, Knots, Weights, Degree, MaxError, MaxDistance : )

Description

gen_contour_nurbs_xld generates an XLD Contour that approximates a NURBS curve (non-uniform rational B-Spline curve). The NURBS curve is specified by its Degree, a control polygon (via Rows and Cols), a Weights vector and a Knots vector.

The Degree of the curve determines the grade of differentiability of the curve. The curve lies within the convex hull of its control polygon (Rows,Cols). The weights in Weights determine how much the curve is attracted by the individual control points (Rows,Cols). A Weights vector with equal weights for all control points is generated if 'auto' is chosen. The Knots vector describes the domain of the curve. Typically, this parameter can be set to 'auto'.

The accuracy of the generated Contour depends on the two parameters MaxDistance and MaxError. MaxDistance limits the maximum distance of two subsequent Contour points. The maximum distance between the generated Contour and the actual NURBS curve is limited to MaxError. By default the approximation must fulfill both constraints, but it is possible to set one of these parameters to 'omit'. The lower MaxError and MaxDistance are chosen the better is the approximation of the curve. Note that all Contour points lie exactly on the curve (except for numerical inaccuracies).

Definition

A NURBS curve C of degree p is given by The basis functions N_{i,p} are defined by Thus, each NURBS curve is determined by its Degree p, its control Polygon P_{0},...,P_{n-1} (Rows and Cols), its weight vector w = (w_{0}, ..., w_{n-1}) (Weights), and its knot vector u = (u_{0},..., u_{n+p}) (Knots).

The grade of differentiability of the curve is set via the degree p. At least p+1 control points P_{i} are required to form a valid curve. The knot vector u describes the domain of the curve. Its length must be n+p+1 and its entries must be monotonically increasing. Every entry u_{i} stands for a knot between the curve segments [u_{i-1}, u_{i}) and [u_{i},u_{i+1}). Each curve segment [u_{i}, u_{i+1}) lies within the convex hull of the control points P_{i-p}, .., P_{i}. For every curve segment of length zero, the differentiability at knot k decreases. The number of occurrences of each knot in the knot vector is called multiplicity. The maximum multiplicity of one knot is p. Only the start and the end point's multiplicity can be p+1 (which in fact is the case for most curves, because in this case start and end point of the curve correspond to the first respectively last point of the control polygon). If start and/or end point do not have full multiplicity, the domain of the curve is restricted to [u_{p},u_{n}]. The weights w_{i} determine the influence of the individual control Points P_{i} on the curve.

Closed Curves: If start and end point are equal (P_{0}=P_{n-1}) and Knots is set to 'auto', the curve is closed automatically such that all knots feature multiplicity one. Note that in this case the length of Weights is required to be n-1.

Execution Information

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Processed without parallelization.

Parameters

Contour (output_object)  xld_cont object

The contour that approximates the NURBS curve.

Rows (input_control)  coordinates.y-array (real)

Row coordinates of the control polygon.

Number of elements: Rows == Cols && Rows >= Degree + 1

Cols (input_control)  coordinates.x-array (real)

Column coordinates of the control polygon.

Number of elements: Cols == Rows && Cols >= Degree + 1

Knots (input_control)  string(-array) (string / real)

The knot vector u.

Default value: 'auto'

Suggested values: 'auto'

Weights (input_control)  string(-array) (string / real)

The weight vector w.

Number of elements: Weights == Rows

Default value: 'auto'

Suggested values: 'auto'

Restriction: Weights > 0.0

Degree (input_control)  integer (integer)

The degree p of the NURBS curve.

Default value: 3

Suggested values: 2, 3, 4, 5

Restriction: Degree >= 2

MaxError (input_control)  real (real / string)

Maximum distance between the NURBS curve and its approximation.

Default value: 1.0

Suggested values: 'omit', 1.0, 2.0, 3.0, 4.0, 5.0

Restriction: MaxError > 0.0

MaxDistance (input_control)  real (real / string)

Maximum distance between two subsequent Contour points.

Default value: 5.0

Suggested values: 'omit', 1.0, 2.0, 3.0, 4.0, 5.0

Restriction: MaxDistance > 0.0

Example (HDevelop)

* use a polygon XLD contour as control polygon and approximate
* the NURBS curve as contour

get_contour_xld(Polygon, Row, Col)
gen_contour_nurbs_xld(NURBSContour, Row, Col, 'auto', 'auto', 3, 1.0, 10.0)

References

L. Piegl, W. Tiller: ”The NURBS Book”, 2nd Edition, Springer, 1997.

Foundation