MbNurbs3D NURBS Curve

The MbNurbs3D class is declared in the cur_nurbs3d.h file.

The NURBS curve takes its name from the initials of the words NonUniform Rational B-Spline. The curve is derived from the MbPolyCurve3D curve. The curve is defined by the set of control points SArray<MbCartPoint3D> pointList, the set of weights of the control points weights, the knot vector knots, the spline order degree, the curve form parameter form, and the curve periodicity flag closed. The curve also has some additional data that are optional and used to speed up the execution of its methods.

The curve is constructed based on B-splines. The knot vector knots is a nondecreasing sequence of real numbers and defines the domain of the curve parameter and the curve shape. The curve form parameter form generally takes the ncf_Unspecified value; in special cases it stores information about the original curve from which the NURBS copy was obtained. The NURBS curve order degree is equal to the order of divided differences used to compute the B-splines. Let the knot vector contain knotsCount elements and the set of control points contain pointsCount elements. For a non-periodic NURBS curve, the numbers of elements in the sets are related by knotsCount = pointsCount + degree. For a periodic NURBS curve, the numbers of elements in the sets are related by knotsCount = pointsCount + 2 degree – 1.

In the PointOn( double & t, MbCartPoint3D & r ) method, the radius vector of the curve r is defined by the vector function

\[r(t) = \frac{\sum\limits_{j = 0}^{\mathit{pointsCount} - 1} {N_{j^{\mathit{degree}}}(t)\, \mathit{weight}[j]\, \mathbf{pointList}[j]}} {\sum\limits_{j = 0}^{\mathit{pointsCount} - 1} {N_{j^{\mathit{degree}}}(t)\, \mathit{weight}[j]}},\]

where \(N_{j}^{\mathit{degree}}(t)\) are B-splines of order degree for the j-th control point pointList[j].

The NURBS curve is shown in Fig. 1:

Fig. 1

The curve parameter t takes values on the interval tmin ≤ t ≤ tmax, where tmin = knots [degree – 1], tmax = knots [knotsCount – degree]. The curve may be periodic. A periodic NURBS curve is shown in Fig. 2:

Fig. 2

The NURBS curve shape depends on the control point locations, the control point weights, and the knot vector values. In the general case, a NURBS curve does not pass through the set of points pointList[i], i = 0,…, pointsCount – 1. For an open NURBS curve to pass through the end control points, the first degree elements and the last degree elements of the knot vector knots must coincide. The larger the weight of a control point, the closer the curve passes to this point, all other conditions being equal.

Each curve can construct its own NURBS copy using the virtual method NurbsCurve( const MbNurbsParameters & tParameters ).