| | Definition of the B_spline curve.
A B-spline curve can be
Uniform or non-uniform
Rational or non-rational
Periodic or non-periodic
a b-spline curve is defined by :
its degree; the degree for a
Geom_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 <br>
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 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
Geom_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, verifying:
- 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.
Warning
In this class, weight value is considered to be zero if
the weight is less than or equal to gp::Resolution().
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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