| | Describes a BSpline surface.
In each parametric direction, a BSpline surface can be:
- uniform or non-uniform,
- rational or non-rational,
- periodic or non-periodic.
A BSpline surface is defined by:
- its degrees, in the u and v parametric directions,
- its periodic characteristic, in the u and v parametric directions,
- a table of poles, also called control points (together
with the associated weights if the surface is rational), and
- a table of knots, together with the associated multiplicities.
The degree of a Geom_BSplineSurface is limited to
a value (25) which is defined and controlled by the
system. This value is returned by the function MaxDegree.
Poles and Weights
Poles and Weights are manipulated using two associative double arrays:
- the poles table, which is a double array of gp_Pnt points, and
- the weights table, which is a double array of reals.
The bounds of the poles and weights arrays are:
- 1 and NbUPoles for the row bounds (provided
that the BSpline surface is not periodic in the u
parametric direction), where NbUPoles is the
number of poles of the surface in the u parametric direction, and
- 1 and NbVPoles for the column bounds (provided
that the BSpline surface is not periodic in the v
parametric direction), where NbVPoles is the
number of poles of the surface in the v parametric direction.
The poles of the surface are the points used to shape
and reshape the surface. They comprise a rectangular network.
If the surface is not periodic:
- The points (1, 1), (NbUPoles, 1), (1,
NbVPoles), and (NbUPoles, NbVPoles)
are the four parametric "corners" of the surface.
- The first column of poles and the last column of
poles define two BSpline curves which delimit the
surface in the v parametric direction. These are the
v isoparametric curves corresponding to the two
bounds of the v parameter.
- The first row of poles and the last row of poles
define two BSpline curves which delimit the surface
in the u parametric direction. These are the u
isoparametric curves corresponding to the two bounds of the u parameter.
If the surface is periodic, these geometric properties are not verified.
It is more difficult to define a geometrical significance
for the weights. However they are useful for
representing a quadric surface precisely. Moreover, if
the weights of all the poles are equal, the surface has
a polynomial equation, and hence is a "non-rational surface".
The non-rational surface is a special, but frequently
used, case, where all poles have identical weights.
The weights are defined and used only in the case of
a rational surface. The rational characteristic is
defined in each parametric direction. A surface can be
rational in the u parametric direction, and
non-rational in the v parametric direction.
Knots and Multiplicities
For a Geom_BSplineSurface the table of knots is
made up of two increasing sequences of reals, without
repetition, one for each parametric direction. The
multiplicities define the repetition of the knots.
A BSpline surface comprises multiple contiguous
patches, which are themselves polynomial or rational
surfaces. The knots are the parameters of the
isoparametric curves which limit these contiguous
patches. The multiplicity of a knot on a BSpline
surface (in a given parametric direction) is related to
the degree of continuity of the surface at that knot in
that parametric direction:
Degree of continuity at knot(i) = Degree - Multi(i) where:
- Degree is the degree of the BSpline surface in
the given parametric direction, and
- Multi(i) is the multiplicity of knot number i in
the given parametric direction.
There are some special cases, where the knots are
regularly spaced in one parametric direction (i.e. the
difference between two consecutive knots is a constant).
- "Uniform": all the multiplicities are equal to 1.
- "Quasi-uniform": all the multiplicities are equal to 1,
except for the first and last knots in this parametric
direction, and these are equal to Degree + 1.
- "Piecewise Bezier": all the multiplicities are equal to
Degree except for the first and last knots, which
are equal to Degree + 1. This surface is a
concatenation of Bezier patches in the given
parametric direction.
If the BSpline surface is not periodic in a given
parametric direction, the bounds of the knots and
multiplicities tables are 1 and NbKnots, where
NbKnots is the number of knots of the BSpline
surface in that parametric direction.
If the BSpline surface is periodic in a given parametric
direction, and there are k periodic knots and p
periodic poles in that parametric direction:
- the period is such that:
period = Knot(k+1) - Knot(1), and
- the poles and knots tables in that parametric
direction can be considered as infinite tables, such that:
Knot(i+k) = Knot(i) + period, and
Pole(i+p) = Pole(i)
Note: The data structure tables for a periodic BSpline
surface are more complex than those of a non-periodic one.
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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