Bernstein-Bézier representation of volumes
Bernstein-Bézier representation of volumes is the description of spatial domains by the method of Bernstein-Bézier. The well known tensor product Bézier representation of surfaces can easily be generalized to volumes by \[ b(u,v,w)=\sum^{l}_{i=0}\sum^{m}_{j=0}\sum^{n}_{k=0}b_{ijk}B\;sp l_ i(u)B^ m_ j(v)B^ n_ k(w),\quad b_{ijk}\in R^ 3,\quad u,v,w\in [0,1], \] where \(B^ l_ i(u)\) are the Bernstein polynomials of degree l in u, etc. The coefficients \(b_{ijk}\in R\times R\times R\) are called Bézier points. In this paper the author studies two problems of the tensor product description of spatial domains by the method of Bernstein-Bézier. First, a volume point b(u,v,w) can be constructed by a generalized construction of the de Casteljau algorithm for Bézier curves; for example, in the u direction using the recursion formula \(b^{\beta \delta \rho}_{\alpha \gamma \epsilon}=(1-u)b_{\alpha \gamma \epsilon}^{\beta -1\delta \rho}+u^{\beta \delta r}_{\alpha +1\gamma \epsilon}\) where \(b^{ijk}_{ijk}=b_{ijk}\) and \(b(u,v,w)=b^{lmn}_{000}\). The first derivative of b(u,v,w) with respect to one variable, say u, is a volume of degree (l-1,m,n). The Bézier point of all higher order derivatives can be constructed by a recurrent method. It's used the following expression \[ \frac{\partial^{p+q+r}}{\partial u^ p\partial v^ q\partial w^ r}b(u,v,w)=\frac{l!}{(l-p)!}\cdot \quad \frac{m!}{(m-q)!}\cdot \frac{n!}{(n-r)!}\Delta^{pqr}b_{000}^{l-pm-qn-\quad r} \] to construct all derivatives by the same de Casteljau scheme. Secondly, the Bézier volume can also be defined by given points \(P_{ijk}\) together with corresponding knots \((u_ i,v_ i,w_ k)\), such that \(b(u_ i,v_ i,w_ k)=P_{ijk}\) for all knots. Due to the tensor product structure or the volume this interpolation problem (the calculation of the Bézier points \(b_{ijk})\), will be solved by applying the well known curve interpolation, computing auxiliary arrays: (1) in u direction, for all j, k, then (2) in v direction, for all i, k and finally (3) in w direction, for all i, j. The author gives two examples very interesting in applied mathematic: a) A contour plot showing five contour surfaces of a four-dimensional surface. It might be a temperature \(T=T(x,y,z)\). b) The motion of a changing surface when coming out of a curved, open tunnel.
scientific article; zbMATH DE number 3924062