For control nets outlining a large class of topological polyhedra, not just tensor-product grids, bi-cubic polyhedral splines form a piecewise polynomial, first-order differentiable space that associates one function with each vertex. Akin to tensor-product splines, the resulting smooth surface approximates the polyhedron. Admissible polyhedral control nets consist of quadrilateral faces in a grid-like layout, star-configuration where n ≠ 4 quadrilateral faces join around an interior vertex, n-gon configurations, where 2n quadrilaterals surround an n-gon, polar configurations where a cone of n triangles meeting at a vertex is surrounded by a ribbon of n quadrilaterals, and three types of T-junctions where two quad-strips merge into one.

The bi-cubic pieces of a polyhedral spline have matching derivatives along their break lines, possibly after a known change of variables. The pieces are represented in Bernstein-Bézier form with coefficients depending linearly on the polyhedral control net, so that evaluation, differentiation, integration, moments, and so on, are no more costly than for standard tensor-product splines. Bi-cubic polyhedral splines can be used both to model geometry and for computing functions on the geometry. Although polyhedral splines do not offer nested refinement by refinement of the control net, polyhedral splines support engineering analysis of curved smooth objects. Coarse nets typically suffice since the splines efficiently model curved features. Algorithm 1032 is a C++ library with input-output example pairs and an IGES output choice.

对于概述一大类拓扑多面体的控制网络，而不仅仅是张量积网格，双三次多面体样条形成分段多项式，一阶可微空间，将一个函数与每个顶点相关联。类似于张量积样条，由此产生的光滑表面近似于多面体。可接受的多面体控制网包括网格状布局中的四边形面、星形配置（其中n ≠ 4 个四边形面围绕一个内部顶点连接）、n边形配置（其中2n 个四边形围绕一个n边形）、极地配置（其中锥形在一个顶点相遇的n 个三角形被n条带包围四边形，以及三种类型的 T 形接头，其中两个四边形条合并为一个。

多面体样条的双三次部分沿其断裂线具有匹配的导数，可能在已知的变量变化之后。这些部分以 Bernstein-Bézier 形式表示，其系数线性取决于多面体控制网络，因此评估、微分、积分、力矩等的成本不比标准张量积样条高。双三次多面体样条既可用于几何建模，也可用于计算几何上的函数。虽然多面体样条不提供通过细化控制网的嵌套细化，但多面体样条支持弯曲光滑物体的工程分析。粗略的网络通常就足够了，因为样条可以有效地模拟弯曲特征。算法 1032 是一个 C++ 库，具有输入输出示例对和 IGES 输出选项。