We consider the construction of a polyhedral Delaunay partition as a limit of the sequence of power diagrams (radical partitions). The dual Voronoi diagram is obtained as a limit of the sequence of weighted Delaunay partitions. The problem is reduced to the construction of two dual convex polyhedra, inscribed and superscribed around a circular paraboloid, as a limit of the sequence of pairs of general dual convex polyhedra. The sequence of primal polyhedra should converge to the superscribed polyhedron and the sequence of the dual polyhedra converges to the inscribed polyhedron. We are interested in the case when the vertices of primal polyhedra can move or merge together, i.e., no new faces are allowed for dual polyhedra. These rules define the transformation of the set of initial spheres into the set of Delaunay spheres using radius variation and sphere movement and elimination. Existence theorems are still unavailable but we suggest a functional measuring the deviation of the convex polyhedron from the one inscribed into the paraboloid. It is the discrete Dirichlet functional for the power function which is a linear interpolant of the vertical distance of the dual vertices from the paraboloid. The functional's absolute minimizer is attained on the constant power field, meaning that the inscribed polyhedron can be obtained by a simple translation. This formulation of the functional for the dual surface is not quadratic since the unknowns are the vertices of the primal polyhedron, hence, the transformation of the set of spheres into Delaunay spheres is not unique. We concentrate on the experimental confirmation of the approach viability and put aside mesh quality problems. The zero value of the gradient of the proposed functional defines a manifold describing the evolution of Delaunay spheres. Hence, Delaunay-Voronoi meshes can be optimized using the manifold as a constraint.

翻译：我们认为,构建一个多光谱三角形分区是动力图序列(激进分区)的极限。 获取双倍的Voronoi 图表是作为加权的德拉乌奈分区序列的极限。 问题被简化为两个双二次锥形圆形圆形圆形圆形圆形, 以圆形双球形圆形圆形圆形圆形圆形圆形圆形圆形的序列。 原始圆形圆形圆形圆形圆形圆形圆形的序列应该聚集到上方圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形( 圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形) 。 双向圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆形圆