NURBS curves and surfaces#
NURBS curves and surfaces extend B-splines with per-control-point weights, making the basis functions rational rather than polynomial and allowing exact representation of conics such as circular arcs. This page defines the rational basis functions and the curve or surface built from them, then states their main properties — including local support, partition of unity, and the convex-hull property — first for curves and then for surfaces. Most of the material follows Piegl and Tiller [1997], Chapter 4.
NURBS curves#
Definition#
A Non-Uniform Rational Basis Spline (NURBS) curve is a parametric curve defined by
where \(p\) is the degree of the curve, the coefficients \(\mathbf P_i\) are called control points, and \(R_{i,p}\) are the rational basis functions given by
in which \(w_i\) are the weights of the control points and \(N_{i,p}\) are B-spline basis functions defined on the nondecreasing knot vector \(U\):
The B-spline basis functions are given by the recursive relation
This recursive relation can produce a \(0/0\) quotient, which is defined to be zero by convention.
Mathematical properties of rational basis functions#
Here is a list of some important properties of the rational basis functions:
The relation \(r=n+p+1\) holds, where \(n+1\) is the number of basis functions and \(r+1\) is the number of elements of the knot vector \(U\).
The numerator and denominator of \(R_{i,p}(u)\) are, at most, polynomials of degree \(p\).
Local support:
\(R_{i,p}(u)=0\) if \(u\) is outside the interval \([u_i,u_{i+p+1})\).
In any given knot interval \([u_{i_0},u_{i_0+1})\), at most \(p+1\) basis functions are nonzero, namely \(R_{i,p}(u)\) with \(i_0-p\leq i\leq i_0\).
Non-negativity: if all weights are positive, \(R_{i,p}(u)\geq0\) for all \(i\), \(p\), and \(u\in[0,1]\). A mixed-sign choice of weights can make the denominator vanish or change sign, and is excluded here.
Partition of unity:
\[\sum_{i=0}^{n} R_{i,p}(u) = 1 \qquad \text{for all } u\in[0,1].\]In addition, for an arbitrary knot span \([u_{i_0},u_{i_0+1})\),
\[\sum_{i=i_0-p}^{i_0} R_{i,p}(u) = 1 \qquad \text{for all } u\in[u_{i_0},u_{i_0+1}).\]This means that the sum of the nonzero basis functions of any knot span is unity.
Continuity and differentiability:
The basis functions are infinitely differentiable in the interior of the knot intervals.
The basis functions are \(p-k\) continuously differentiable at a knot with multiplicity \(k\), for \(1\leq k\leq p\), provided the weighted denominator does not vanish there.
Extrema: for \(p\geq1\) and positive weights, \(R_{i,p}(u)\) attains exactly one maximum in \(u\in[0,1]\).
Unlike the polynomial basis, there is no compact closed-form expression for the derivative of \(R_{i,p}(u)\) purely in terms of other rational basis functions and knots; it also involves the weighted denominator \(w(u)=\sum_kN_{k,p}(u)w_k\) and its derivatives. A usable recursive formula can still be obtained by differentiating \(w(u)R_{i,p}(u)=w_iN_{i,p}(u)\) with the product rule \(k\) times and solving for the highest-order term:
(3)#\[R_{i,p}^{(k)}(u) = \frac{1}{w(u)}\left[ w_i N_{i,p}^{(k)}(u) - \sum_{j=1}^{k}\binom{k}{j} w^{(j)}(u)\, R_{i,p}^{(k-j)}(u) \right],\]where \(w^{(j)}=\sum_kN_{k,p}^{(j)}(u)w_k\). This is the same Leibniz-rule technique used below for the coordinates of a NURBS curve, applied here to a single basis function instead of to \(\mathbf C(u)\).
When the first and last knots have multiplicity \(p+1\), the knot vector is a clamped knot vector as in the B-spline case, and basis functions of clamped knot vectors satisfy two additional properties:
\(R_{0,p}(u=0)=1\) and \(R_{i,p}(u=0)=0\) for \(i\neq0\).
\(R_{n,p}(u=1)=1\) and \(R_{i,p}(u=1)=0\) for \(i\neq n\).
If all the control-point weights are equal and nonzero, the rational basis functions reduce to B-spline basis functions, that is, \(R_{i,p}(u)=N_{i,p}(u)\).
If all the control-point weights are equal and nonzero, the knot vector is clamped, and \(p=n\), then the rational basis functions reduce to Bernstein polynomials, that is, \(R_{i,p}(u)=B_{i,n}(u)\).
Mathematical properties of NURBS curves#
Here is a list of some important properties of NURBS curves:
\(\mathbf C(u)\) is a piecewise curve, and its components are ratios of polynomials of, at most, degree \(p\).
The degree \(p\), number of control points \(n+1\), and number of knots \(r+1\) are related by \(r=n+p+1\).
NURBS curves contain B-spline and rational/non-rational Bézier curves as special cases:
If all the control-point weights are equal, the NURBS curve reduces to a B-spline curve.
If \(n=p\) and the knot vector is clamped, the NURBS curve reduces to a rational Bézier curve.
If all the control-point weights are equal, \(p=n\), and the knot vector is clamped, the NURBS curve reduces to a polynomial Bézier curve.
Affine invariance: NURBS curves are invariant under affine transformations such as rotations, displacements, and scalings. One can apply an affine transformation to the curve by applying it to its control points. Given an affine transformation \(\phi(\mathbf v)=A\mathbf v+\mathbf b\),
\[\phi\big(\mathbf C(u)\big) = \phi\Big(\sum_{i=0}^{n} R_{i,p}(u)\,\mathbf P_i\Big) = A\sum_{i=0}^{n} R_{i,p}(u)\,\mathbf P_i + \mathbf b = A\sum_{i=0}^{n} R_{i,p}(u)\,\mathbf P_i + \mathbf b\sum_{i=0}^{n} R_{i,p}(u)\]\[= \sum_{i=0}^{n} R_{i,p}(u)\,(A\mathbf P_i+\mathbf b) = \sum_{i=0}^{n} R_{i,p}(u)\,\phi(\mathbf P_i),\]using the partition of unity in the middle step.
Convex hull property: with positive weights, all the points of a NURBS curve are contained in the convex hull of its control points,
\[\mathcal{CH}(P) = \Big\{\, \mathbf C = \sum_{k=0}^{N} a_k\, \mathbf P_k \ \text{such that}\ \sum_{k=0}^{N} a_k = 1 \text{ and } a_k\geq0 \text{ for } k=0,1,\ldots,N \,\Big\}.\]With mixed-sign weights this can fail: a two-point rational curve can evaluate outside the segment joining its control points.
Strong convex hull property: if \(u\in[u_{i_0},u_{i_0+1})\), then \(\mathbf C(u)\) is contained in the convex hull of the control points \(\mathbf P_i\) with \(i_0-p\leq i\leq i_0\).
Local modification scheme: modifying the control point \(\mathbf P_i\) or weight \(w_i\) affects \(\mathbf C(u)\) only on the interval \([u_i,u_{i+p+1})\). This follows from \(R_{i,p}(u)=0\) outside that interval, and it implies that the shape of a NURBS curve can be modified locally without changing its shape globally.
The polygon formed by the set of control points is known as the control polygon. The control polygon represents a piecewise linear approximation to the NURBS curve.
Variation diminishing property: with positive weights, no straight line (or plane in three dimensions) intersects the NURBS curve more times than it intersects its control polygon. Intuitively, the curve does not wiggle more than its control polygon.
Continuity and differentiability:
NURBS curves are infinitely differentiable in the interior of the knot intervals, wherever the weighted denominator does not vanish.
NURBS curves are at least \(p-k\) continuously differentiable at a knot with multiplicity \(k\).
Homogeneous coordinates: \(N\)-dimensional rational functions with a common denominator can be represented as polynomial functions in \((N+1)\)-dimensional space using homogeneous coordinates.
Consider a three-dimensional point \(\mathbf P=(x,y,z)\). It can be written in four-dimensional space as \(\mathbf P^w=(wx,wy,wz,w)=(X,Y,Z,W)\), where \(w\neq0\). The point \(\mathbf P\) is recovered from \(\mathbf P^w\) by the mapping \(\mathcal H\),
\[\mathbf P = \mathcal H\{\mathbf P^w\} = \mathcal H\{(X,Y,Z,W)\} = \left(\frac{X}{W}, \frac{Y}{W}, \frac{Z}{W}\right).\]This lets an \(N\)-dimensional NURBS curve be represented as an \((N+1)\)-dimensional B-spline curve, whose coordinates are then mapped back with \(\mathcal H\). Given a NURBS curve \(\mathbf C(u)\) with control points \(\mathbf P_i\) and weights \(w_i\), construct the weighted control points
(4)#\[\mathbf P_i^w = (w_ix_i,\, w_iy_i,\, w_iz_i,\, w_i)\]and define the corresponding non-rational B-spline curve in four-dimensional space,
\[\mathbf C^w(u) = \sum_{i=0}^{n} N_{i,p}(u)\, \mathbf P_i^w, \qquad 0\leq u\leq1.\]The coordinates of the original NURBS curve are recovered using standard B-spline algorithms to evaluate \(\mathbf C^w(u)\) and then applying \(\mathcal H\):
\[\mathbf C(u) = \mathcal H\{\mathbf C^w(u)\} = \mathcal H\Big\{\sum_{i=0}^{n} N_{i,p}(u)\,\mathbf P_i^w\Big\} = \frac{\sum\limits_{i=0}^{n} N_{i,p}(u)\,w_i\,\mathbf P_i} {\sum\limits_{i=0}^{n} N_{i,p}(u)\,w_i} = \sum_{i=0}^{n} R_{i,p}(u)\,\mathbf P_i.\]This property is the key to evaluating the derivatives of NURBS curves.
First and higher order derivatives:
The derivatives of a NURBS curve \(\mathbf C(u)\) can be expressed in terms of the derivatives of the corresponding B-spline curve in homogeneous space, \(\mathbf C^w(u)\). Write
\[\mathbf C(u) = \frac{\sum\limits_{i=0}^{n} N_{i,p}(u)\,w_i\,\mathbf P_i} {\sum\limits_{i=0}^{n} N_{i,p}(u)\,w_i} = \frac{\mathbf A(u)}{w(u)},\]where \(\mathbf A(u)\) collects the first three coordinates of \(\mathbf C^w(u)\) and \(w(u)\) is its fourth coordinate. To compute the first derivative, clear the denominator, differentiate, and solve for \(\mathbf C'(u)\):
\[w\,\mathbf C = \mathbf A \ \Longrightarrow\ w'\,\mathbf C + w\,\mathbf C' = \mathbf A' \ \Longrightarrow\ \mathbf C' = \frac{1}{w}\big(\mathbf A' - w'\,\mathbf C\big).\]To compute the \(k\)-th order derivative, differentiate \(k\) times using Leibniz’ rule for products, then solve for \(\mathbf C^{(k)}(u)\):
(5)#\[\big[w\,\mathbf C\big]^{(k)} = \mathbf A^{(k)} \ \Longrightarrow\ \sum_{i=0}^{k}\binom{k}{i} w^{(i)}\,\mathbf C^{(k-i)} = \mathbf A^{(k)} \ \Longrightarrow\ \mathbf C^{(k)} = \frac{1}{w}\left( \mathbf A^{(k)} - \sum_{i=1}^{k}\binom{k}{i} w^{(i)}\,\mathbf C^{(k-i)} \right).\]The derivatives of \(\mathbf A(u)\) and \(w(u)\) are obtained directly by differentiating \(\mathbf C^w(u)\), which is a polynomial B-spline curve:
\[\big[\mathbf C^w\big]^{(k)}(u) = \sum_{i=0}^{n} N_{i,p}^{(k)}(u)\, \mathbf P_i^w.\]A related, useful fact is how the curve responds to a change in a single weight: differentiating (1) with respect to \(w_i\) using (2) gives
\[\frac{\partial \mathbf C}{\partial w_i} = \frac{N_{i,p}(u)}{w(u)}\big(\mathbf P_i - \mathbf C(u)\big),\]so increasing \(w_i\) pulls the curve toward \(\mathbf P_i\) on the support of \(N_{i,p}\), and rescaling every weight by the same nonzero constant leaves the curve unchanged.
Endpoint interpolation: the start and end points of a clamped NURBS curve coincide with the first and last control points, respectively:
\[\mathbf C(u=0) = \mathbf P_0, \qquad \mathbf C(u=1) = \mathbf P_n.\]Endpoint tangency: for \(p\geq1\), a clamped NURBS curve is tangent to the control polygon at the endpoints,
\[\frac{\mathrm d\mathbf C}{\mathrm du}\Big|_{u=0} = \left(\frac{p}{u_{p+1}}\right)\left(\frac{w_1}{w_0}\right) (\mathbf P_1-\mathbf P_0), \qquad \frac{\mathrm d\mathbf C}{\mathrm du}\Big|_{u=1} = \left(\frac{p}{1-u_n}\right)\left(\frac{w_{n-1}}{w_n}\right) (\mathbf P_n-\mathbf P_{n-1}).\]Endpoint curvature: for \(p\geq2\), the second derivative of a clamped NURBS curve at its endpoints is given by
\[\frac{\mathrm d^2\mathbf C}{\mathrm du^2}\Big|_{u=0} = \frac{p(p-1)}{u_{p+1}}\left[ \left(\frac{1}{u_{p+2}}\right)\left(\frac{w_2}{w_0}\right)(\mathbf P_2-\mathbf P_0) - \left(\frac{1}{u_{p+1}}+\frac{1}{u_{p+2}}\right)\left(\frac{w_1}{w_0}\right)(\mathbf P_1-\mathbf P_0) \right]\]\[{}+ \frac{2p^2}{u_{p+1}^2}\left(\frac{w_1}{w_0}\right)\left(1-\frac{w_1}{w_0}\right)(\mathbf P_1-\mathbf P_0),\]\[\frac{\mathrm d^2\mathbf C}{\mathrm du^2}\Big|_{u=1} = \frac{p(p-1)}{1-u_n}\left[ \left(\frac{1}{1-u_{n-1}}\right)\left(\frac{w_{n-2}}{w_n}\right)(\mathbf P_{n-2}-\mathbf P_n) - \left(\frac{1}{1-u_n}+\frac{1}{1-u_{n-1}}\right)\left(\frac{w_{n-1}}{w_n}\right)(\mathbf P_{n-1}-\mathbf P_n) \right]\]\[{}+ \frac{2p^2}{(1-u_n)^2}\left(\frac{w_{n-1}}{w_n}\right)\left(1-\frac{w_{n-1}}{w_n}\right)(\mathbf P_{n-1}-\mathbf P_n).\]Provided the endpoint control-polygon edge is nonzero, the curvature of a clamped NURBS curve at its endpoints is given by
(6)#\[\kappa(u=0) = \left(\frac{p-1}{p}\right)\left(\frac{u_{p+1}}{u_{p+2}}\right) \left(\frac{w_0w_2}{w_1^2}\right) \frac{\big\|(\mathbf P_2-\mathbf P_0)\times(\mathbf P_1-\mathbf P_0)\big\|} {\|\mathbf P_1-\mathbf P_0\|^3},\]\[\kappa(u=1) = \left(\frac{p-1}{p}\right)\left(\frac{1-u_n}{1-u_{n-1}}\right) \left(\frac{w_nw_{n-2}}{w_{n-1}^2}\right) \frac{\big\|(\mathbf P_{n-2}-\mathbf P_n)\times(\mathbf P_{n-1}-\mathbf P_n)\big\|} {\|\mathbf P_{n-1}-\mathbf P_n\|^3}.\]These formulas require \(p\geq2\) and a nonzero endpoint tangent. They reduce to the B-spline endpoint curvature formulas (7) when the endpoint weight ratios are one, and further to the Bézier formulas (7) under the additional B-spline special cases above. See G² continuity for how these formulas are used to join curve segments smoothly.
NURBS surfaces#
Definition#
A Non-Uniform Rational Basis Spline (NURBS) surface is a parametric surface defined by
where \(p\) and \(q\) are the degrees of the surface in the \(u\)- and \(v\)-directions, the coefficients \(\mathbf P_{i,j}\) are a bidirectional net of control points, and \(R_{i,j}^{p,q}(u,v)\) are the rational basis functions given by
in which \(w_{i,j}\) are the weights of the control points and \(N_{i,p}(u)N_{j,q}(v)\) is the product of univariate B-spline basis functions defined on the nondecreasing knot vectors \(U\) and \(V\):
The denominator sums over both the \(u\)-degree basis \(N_{a,p}(u)\) and the \(v\)-degree basis \(N_{b,q}(v)\); the \(v\)-direction factor must use the degree \(q\), not \(p\), since \(p\) and \(q\) can differ. The \(u\)-direction basis functions are given by the recursive relation of the B-spline case, while the \(v\)-direction basis functions are defined analogously, replacing the variable \(u\) by \(v\) and the indices \(i\) and \(p\) by \(j\) and \(q\), respectively.
Mathematical properties of bivariate rational basis functions#
Here is a list of some important properties of the bivariate rational basis functions:
The relation \(r=n+p+1\) holds, where \(n+1\) is the number of basis functions in the \(u\)-direction and \(r+1\) is the number of elements of the knot vector \(U\). Likewise, \(s=m+q+1\) relates \(m+1\) and the \(s+1\) elements of \(V\).
The numerator and denominator of \(R_{i,j}^{p,q}(u,v)\) are, at most, polynomials of bidegree \((p,q)\).
Local support:
\(R_{i,j}^{p,q}(u,v)=0\) if \((u,v)\) is outside the rectangle \([u_i,u_{i+p+1})\times[v_j,v_{j+q+1})\).
In any given knot rectangle \([u_{i_0},u_{i_0+1})\times[v_{j_0},v_{j_0+1})\), at most \((p+1)(q+1)\) basis functions are nonzero, namely \(R_{i,j}^{p,q}(u,v)\) with \(i_0-p\leq i\leq i_0\) and \(j_0-q\leq j\leq j_0\).
Non-negativity: with positive weights, \(R_{i,j}^{p,q}(u,v)\geq0\) for all \(i\), \(j\), \(p\), \(q\), and \((u,v)\in[0,1]\times[0,1]\).
Partition of unity:
\[\sum_{i=0}^{n}\sum_{j=0}^{m} R_{i,j}^{p,q}(u,v) = 1 \qquad \text{for all } (u,v)\in[0,1]\times[0,1].\]In addition, for an arbitrary knot rectangle \([u_{i_0},u_{i_0+1})\times[v_{j_0},v_{j_0+1})\),
\[\sum_{i=i_0-p}^{i_0}\sum_{j=j_0-q}^{j_0} R_{i,j}^{p,q}(u,v) = 1 \qquad \text{for all } (u,v)\in[u_{i_0},u_{i_0+1})\times[v_{j_0},v_{j_0+1}).\]This means that the sum of the nonzero basis functions of any knot rectangle is unity.
Continuity and differentiability:
The basis functions are infinitely differentiable in the interior of the knot rectangles formed by \(U\) and \(V\).
The basis functions are \(p-k\) (respectively \(q-k\)) continuously differentiable in the \(u\)-direction (respectively \(v\)-direction) at a \(u\)-knot (respectively \(v\)-knot) with multiplicity \(k\).
Extrema: for \(p,q\geq1\) and positive weights, \(R_{i,j}^{p,q}(u,v)\) attains exactly one maximum in \((u,v)\in[0,1]\times[0,1]\).
As in the curve case, there is no compact closed-form expression for the partial derivatives of \(R_{i,j}^{p,q}(u,v)\) purely in terms of other rational basis functions and knots. They can, however, be obtained with the same Leibniz-rule technique used below for the coordinates of a NURBS surface, applied to a single basis function instead of to \(\mathbf S(u,v)\).
If all the control-point weights are equal, the rational basis functions reduce to B-spline basis functions, that is, \(R_{i,j}^{p,q}(u,v)=N_{i,p}(u)\,N_{j,q}(v)\).
If all the control-point weights are equal, the knot vectors are clamped, and \((p,q)=(n,m)\), then the rational basis functions reduce to Bernstein polynomials, that is, \(R_{i,j}^{p,q}(u,v)=B_{i,n}(u)\,B_{j,m}(v)\).
Mathematical properties of NURBS surfaces#
Here is a list of some important properties of NURBS surfaces:
\(\mathbf S(u,v)\) is a piecewise surface, and its components are ratios of bivariate polynomials of, at most, bidegree \((p,q)\).
In the \(u\)-direction, the degree \(p\), number of control points \(n+1\), and number of knots \(r+1\) are related by \(r=n+p+1\). In the \(v\)-direction, the degree \(q\), number of control points \(m+1\), and number of knots \(s+1\) are related by \(s=m+q+1\).
NURBS surfaces contain B-spline and rational/non-rational Bézier surfaces as special cases:
If all the control-point weights are equal, the NURBS surface reduces to a B-spline surface.
If \((p,q)=(n,m)\) and the knot vectors are clamped, the NURBS surface reduces to a rational Bézier surface.
If all the control-point weights are equal, \((p,q)=(n,m)\), and the knot vectors are clamped, the NURBS surface reduces to a polynomial Bézier surface.
Affine invariance: NURBS surfaces are invariant under affine transformations such as rotations, displacements, and scalings. Given an affine transformation \(\phi(\mathbf v)=A\mathbf v+\mathbf b\),
\[\phi\big(\mathbf S(u,v)\big) = \phi\Big(\sum_{i,j} R_{i,j}^{p,q}(u,v)\,\mathbf P_{i,j}\Big) = A\sum_{i,j} R_{i,j}^{p,q}(u,v)\,\mathbf P_{i,j} + \mathbf b\sum_{i,j} R_{i,j}^{p,q}(u,v)\]\[= \sum_{i,j} R_{i,j}^{p,q}(u,v)\,(A\mathbf P_{i,j}+\mathbf b) = \sum_{i,j} R_{i,j}^{p,q}(u,v)\,\phi(\mathbf P_{i,j}),\]using the partition of unity in the middle step.
Convex hull property: with positive weights, all the points of a NURBS surface are contained in the convex hull of its control points, defined as in the curve case.
Strong convex hull property: if \((u,v)\in[u_{i_0},u_{i_0+1})\times[v_{j_0},v_{j_0+1})\), then \(\mathbf S(u,v)\) is contained in the convex hull of the control points \(\mathbf P_{i,j}\) with \(i_0-p\leq i\leq i_0\) and \(j_0-q\leq j\leq j_0\).
Local modification scheme: modifying the control point \(\mathbf P_{i,j}\) or weight \(w_{i,j}\) affects \(\mathbf S(u,v)\) only on the rectangle \([u_i,u_{i+p+1})\times[v_j,v_{j+q+1})\). This follows from \(R_{i,j}^{p,q}(u,v)=0\) outside that rectangle, and it implies that the shape of a NURBS surface can be modified locally without changing its shape globally.
If triangulated, the net of control points represents a piecewise planar approximation to the NURBS surface.
No known variation-diminishing property.
Continuity and differentiability:
NURBS surfaces are infinitely differentiable in the interior of the knot rectangles formed by \(U\) and \(V\), wherever the weighted denominator does not vanish.
NURBS surfaces are \(p-k\) (respectively \(q-k\)) continuously differentiable in the \(u\)-direction (respectively \(v\)-direction) at a \(u\)-knot (respectively \(v\)-knot) with multiplicity \(k\).
Homogeneous coordinates: as for curves, an \(N\)-dimensional rational surface can be represented as an \((N+1)\)-dimensional polynomial B-spline surface using homogeneous coordinates, and the result mapped back with \(\mathcal H\).
Given a NURBS surface \(\mathbf S(u,v)\) with control points \(\mathbf P_{i,j}\) and weights \(w_{i,j}\), construct the weighted control points \(\mathbf P_{i,j}^w=(w_{i,j}\mathbf P_{i,j},\,w_{i,j})\), that is, in three spatial dimensions,
(9)#\[\mathbf P_{i,j}^w = (w_{i,j}x_{i,j},\, w_{i,j}y_{i,j},\, w_{i,j}z_{i,j},\, w_{i,j}),\]and define the corresponding non-rational B-spline surface in four-dimensional space, using the polynomial tensor-product basis, not the rational one:
(10)#\[\mathbf S^w(u,v) = \sum_{i=0}^{n}\sum_{j=0}^{m} N_{i,p}(u)\,N_{j,q}(v)\, \mathbf P_{i,j}^w, \qquad 0\leq(u,v)\leq1.\]The coordinates of the original NURBS surface are recovered using standard B-spline algorithms to evaluate \(\mathbf S^w(u,v)\) and then applying \(\mathcal H\):
\[\mathbf S(u,v) = \mathcal H\{\mathbf S^w(u,v)\} = \frac{\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{m} N_{i,p}(u)\,N_{j,q}(v)\,w_{i,j}\,\mathbf P_{i,j}} {\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{m} N_{i,p}(u)\,N_{j,q}(v)\,w_{i,j}} = \sum_{i=0}^{n}\sum_{j=0}^{m} R_{i,j}^{p,q}(u,v)\,\mathbf P_{i,j}.\]This property is the key to evaluating the derivatives of NURBS surfaces.
First and higher order derivatives:
The derivatives of a NURBS surface \(\mathbf S(u,v)\) can be expressed in terms of the derivatives of the corresponding B-spline surface in homogeneous space, \(\mathbf S^w(u,v)\). Write
\[\mathbf S(u,v) = \frac{\sum\limits_{i,j} N_{i,p}(u)\,N_{j,q}(v)\,w_{i,j}\,\mathbf P_{i,j}} {\sum\limits_{i,j} N_{i,p}(u)\,N_{j,q}(v)\,w_{i,j}} = \frac{\mathbf A(u,v)}{w(u,v)},\]where \(\mathbf A(u,v)\) collects the first three coordinates of \(\mathbf S^w(u,v)\) and \(w(u,v)\) is its fourth coordinate. To compute a first partial derivative with respect to \(\alpha\in\{u,v\}\), clear the denominator, differentiate, and solve for \(\mathbf S_\alpha\):
\[w\,\mathbf S = \mathbf A \ \Longrightarrow\ w_\alpha\,\mathbf S + w\,\mathbf S_\alpha = \mathbf A_\alpha \ \Longrightarrow\ \frac{\partial\mathbf S}{\partial\alpha} = \mathbf S_\alpha = \frac{1}{w}\big(\mathbf A_\alpha - w_\alpha\,\mathbf S\big).\]To compute the \((k,l)\)-th order partial derivative, differentiate \(k\) times in \(u\) and \(l\) times in \(v\) using Leibniz’ rule for products, then solve for \(\mathbf S^{(k,l)}(u,v)\):
\[\big[w\,\mathbf S\big]^{(k,l)} = \mathbf A^{(k,l)} \ \Longrightarrow\ \sum_{a=0}^{k}\sum_{b=0}^{l} \binom{k}{a}\binom{l}{b}\, w^{(a,b)}\,\mathbf S^{(k-a,l-b)} = \mathbf A^{(k,l)}.\]Separating the \((a,b)=(0,0)\) term from the rest and solving for \(\mathbf S^{(k,l)}\) gives
(11)#\[\begin{split}\mathbf S^{(k,l)} = \frac{1}{w}\left[ \mathbf A^{(k,l)} - \sum_{\substack{0\leq a\leq k,\ 0\leq b\leq l\\ (a,b)\neq(0,0)}} \binom{k}{a}\binom{l}{b}\, w^{(a,b)}\,\mathbf S^{(k-a,l-b)} \right].\end{split}\]Splitting the excluded index set into \(a>0,\,b=0\); \(a=0,\,b>0\); and \(a>0,\,b>0\) recovers the three separate correction sums used for pure-\(u\), pure-\(v\), and mixed terms, respectively; the single sum above is only a more compact way to write the same correction. The derivatives of \(\mathbf A(u,v)\) and \(w(u,v)\) are obtained directly by differentiating \(\mathbf S^w(u,v)\), which is a polynomial B-spline surface:
\[\big[\mathbf S^w\big]^{(k,l)}(u,v) = \sum_{i=0}^{n}\sum_{j=0}^{m} N_{i,p}^{(k)}(u)\,N_{j,q}^{(l)}(v)\, \mathbf P_{i,j}^w.\]Corner point interpolation: the corners of a clamped NURBS surface coincide with the corner points of its control net,
\[\mathbf S(u=0,v=0) = \mathbf P_{0,0}, \qquad \mathbf S(u=1,v=0) = \mathbf P_{n,0}, \qquad \mathbf S(u=0,v=1) = \mathbf P_{0,m}, \qquad \mathbf S(u=1,v=1) = \mathbf P_{n,m}.\]More generally, each boundary row or column of the control net, together with its weights, defines a rational boundary curve of the surface.