| | Describes a BSpline curve.
A BSpline curve can be:
- uniform or non-uniform,
- rational or non-rational,
- periodic or non-periodic.
A BSpline curve is defined by:
- its degree; the degree for a
Geom2d_BSplineCurve is limited to a value (25)
which is defined and controlled by the system. This
value is returned by the function MaxDegree;
- its periodic or non-periodic nature;
- a table of poles (also called control points), with
their associated weights if the BSpline curve is
rational. The poles of the curve are "control points"
used to deform the curve. If the curve is
non-periodic, the first pole is the start point of the
curve, and the last pole is the end point of the
curve. The segment, which joins the first pole to the
second pole, is the tangent to the curve at its start
point, and the segment, which joins the last pole to
the second-from-last pole, is the tangent to the
curve at its end point. If the curve is periodic, these
geometric properties are not verified. It is more
difficult to give a geometric signification to the
weights but they are useful for providing exact
representations of the arcs of a circle or ellipse.
Moreover, if the weights of all the poles are equal,
the curve has a polynomial equation; it is
therefore a non-rational curve.
- a table of knots with their multiplicities. For a
Geom2d_BSplineCurve, the table of knots is an
increasing sequence of reals without repetition; the
multiplicities define the repetition of the knots. A
BSpline curve is a piecewise polynomial or rational
curve. The knots are the parameters of junction
points between two pieces. The multiplicity
Mult(i) of the knot Knot(i) of the BSpline
curve is related to the degree of continuity of the
curve at the knot Knot(i), which is equal to
Degree - Mult(i) where Degree is the
degree of the BSpline curve.
If the knots are regularly spaced (i.e. the difference
between two consecutive knots is a constant), three
specific and frequently used cases of knot distribution
can be identified:
- "uniform" if all multiplicities are equal to 1,
- "quasi-uniform" if all multiplicities are equal to 1,
except the first and the last knot which have a
multiplicity of Degree + 1, where Degree is
the degree of the BSpline curve,
- "Piecewise Bezier" if all multiplicities are equal to
Degree except the first and last knot which have
a multiplicity of Degree + 1, where Degree is
the degree of the BSpline curve. A curve of this
type is a concatenation of arcs of Bezier curves.
If the BSpline curve is not periodic:
- the bounds of the Poles and Weights tables are 1
and NbPoles, where NbPoles is the number of
poles of the BSpline curve,
- the bounds of the Knots and Multiplicities tables are
1 and NbKnots, where NbKnots is the number
of knots of the BSpline curve.
If the BSpline curve is periodic, and if there are k
periodic knots and p periodic poles, the period is:
period = Knot(k + 1) - Knot(1)
and the poles and knots tables can be considered as
infinite tables, such that:
- Knot(i+k) = Knot(i) + period
- Pole(i+p) = Pole(i)
Note: data structures of a periodic BSpline curve are
more complex than those of a non-periodic one.
Warnings :
In this class we consider that a weight value is zero if
Weight <= Resolution from package gp.
For two parametric values (or two knot values) U1, U2 we
consider that U1 = U2 if Abs (U2 - U1) <= Epsilon (U1).
For two weights values W1, W2 we consider that W1 = W2 if
Abs (W2 - W1) <= Epsilon (W1). The method Epsilon is
defined in the class Real from package Standard.
References :
. A survey of curve and surface methods in CADG Wolfgang BOHM
CAGD 1 (1984)
. On de Boor-like algorithms and blossoming Wolfgang BOEHM
cagd 5 (1988)
. Blossoming and knot insertion algorithms for B-spline curves
Ronald N. GOLDMAN
. Modelisation des surfaces en CAO, Henri GIAUME Peugeot SA
. Curves and Surfaces for Computer Aided Geometric Design,
a practical guide Gerald Farin
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