We give a full classification of continuous flexible discrete axial cone-nets, which are called axial C-hedra. The obtained result can also be used to construct their semi-discrete analogs. Moreover, we identify a novel subclass within the determined class of (semi-)discrete axial cone-nets, whose members are named axial P-nets as they fulfill the proportion (P) of the intercept theorem. Known special cases of these axial P-nets are the smooth and discrete conic crease patterns with reflecting rule lines. By using a parallelism operation one can even generalize axial P-nets. The resulting general P-nets constitute a rich novel class of continuous flexible (semi-)discrete surfaces, which allow direct access to their spatial shapes by three control polylines. This intuitive method makes them suitable for transformable design tasks using interactive tools.

翻译：本文对连续可变形离散轴向锥面网格（称为轴向C-多面体）进行了完整分类，所得结果还可用于构造其半离散模拟。此外，我们在所确定的（半）离散轴向锥面网格类别中识别出一个新子类，其成员因满足截距定理的比例（P）性质而被称为轴向P-网格。这些轴向P-网格的已知特例是光滑及离散的锥形折痕图案，其中包含反射规则线。通过并行性操作可进一步推广轴向P-网格，由此产生的广义P-网格构成了一类丰富新颖的连续可变形（半）离散曲面，可通过三条控制多段线直接获取其空间形态。这种直观方法使其适用于采用交互工具的可变形设计任务。