Surfaces
Based on both bases surfaces are defined.
Definitions
B-Spline Surfaces
A tensor product surface [1, p. 100]
\[\bm{s}(u,v) = \sum_{i=1}^{N_u} \sum_{j=1}^{N_v} B_{i,p}(u) B_{j,q}(v) \bm{p}_{i,j}\]
is defined by introducing two knot-vectors for the B-splines of degree $p$ and $q$ and a second parametric value $v \in [0, 1]$, as well as a net of constant controlpoints $\bm{p}_{i,j} \in \mathbb{R}^3$.
NURBS Surfaces
Analogously, a NURBS surface
\[\bm{s}(u,v) = \sum_{i=1}^{N_u} \sum_{j=1}^{N_v} R_{i,j}(u,v) \bm{p}_{i,j} = \cfrac{\sum_{i=1}^{N_u} \sum_{j=1}^{N_v} B_{i,p}(u) B_{j,q}(v) w_{i,j} \bm{p}_{i,j}}{\sum_{i=1}^{N_u} \sum_{j=1}^{N_v} B_{i,p}(u) B_{j,q}(v) w_{i,j}}\]
is defined.
$\bm{s}(u,v): (u, v) \mapsto (x,y,z)$, that is, the surface is a mapping from the parametric space $(u, v) \in {[0,1]}^2$ to the physical space $\mathbb{R}^3$.
Derivatives
B-Spline Surfaces
The $m$-th derivative in $u$ and $m$-th derivative in $v$ of a B-spline surface is given as [1, p. 111]
\[\bm{s}^{(m,n)}(u,v) = \cfrac{\partial^{m+n}}{\partial^m u \partial^n v}\bm{s}(u,v) = \sum_{i=1}^{N_u} \sum_{j=1}^{N_v} B_{i,p}^{(m)}(u) B_{j,p}^{(n)}(u) \bm{p}_{i,j} \,.\]
NURBS Surfaces
The $m$-th derivative in $u$ and $m$-th derivative in $v$ of a NURBS surface is given as
\[\bm{s}^{(m,n)}(u,v) = \cfrac{\partial^{m+n}}{\partial^m u \partial^n v}\bm{s}(u,v) = \sum_{i=1}^{N_u} \sum_{j=1}^{N_v} R_{i,j}^{(m,n)}(u,v) \bm{p}_{i,j} \,.\]
It can be computed as [1, p. 136]
\[\begin{aligned} \bm{s}^{(m,n)}(u,v) = \cfrac{1}{w^{(0,0)}(u,v)} \Bigg( \bm{a}^{(m,n)}(u,v) \\ &- \sum_{i=1}^m \begin{pmatrix} m \\ i \end{pmatrix} w^{(i,0)}(u,v) \bm{s}^{(m-i, n)}(u,v)\\ &- \sum_{j=1}^n \begin{pmatrix} n \\ j \end{pmatrix} w^{(0,j)}(u,v) \bm{s}^{(m, n-j)}(u,v)\\ &- \sum_{i=1}^m \begin{pmatrix} m \\ i \end{pmatrix} \sum_{j=1}^n \begin{pmatrix} n \\ j \end{pmatrix} w^{(i,j)}(u,v) \bm{s}^{(m-i, n-j)}(u,v) \Bigg) \end{aligned}\]
with
\[\bm{a}^{(m,n)}(u,v) = \sum_{i=1}^{N_u} \sum_{j=1}^{N_v} B_{i,p}^{(m)}(u) B_{j,q}^{(n)}(v) w_{i,j} \bm{p}_{i,j} \]
and
\[w^{(m,n)}(u,v) = \sum_{i=1}^{N_u} \sum_{j=1}^{N_v} B_{i,p}^{(m)}(u) B_{j,q}^{(n)}(v) w_{i,j} \,.\]
Jacobian
The Jacobian matrix $\bm{J}(u,v)$ for the surface is given as
\[\bm{J}(u,v) = \begin{bmatrix} \cfrac{\partial x}{\partial u} & \cfrac{\partial x}{\partial v} \\[3mm] \cfrac{\partial y}{\partial u} & \cfrac{\partial y}{\partial v} \\[3mm] \cfrac{\partial z}{\partial u} & \cfrac{\partial z}{\partial v} \end{bmatrix} = \begin{bmatrix} \bm{s}^{(1, 0)} & \bm{s}^{(0, 1)} \end{bmatrix}\]
and the magnitude of its determinant as
\[\left|\det\left(\bm{J}(u,v)\right)\right| = \sqrt{ {\left( \cfrac{\partial y}{\partial u}\cfrac{\partial z}{\partial v} - \cfrac{\partial z}{\partial u} \cfrac{\partial y}{\partial v} \right)}^2 + {\left( \cfrac{\partial z}{\partial u}\cfrac{\partial x}{\partial v} - \cfrac{\partial x}{\partial u} \cfrac{\partial z}{\partial v} \right)}^2 + {\left( \cfrac{\partial x}{\partial u}\cfrac{\partial y}{\partial v} - \cfrac{\partial y}{\partial u} \cfrac{\partial x}{\partial v} \right)}^2 } \,.\]