From axial C-hedra to general P-nets

We give a full classification of continuous flexible discrete axialcone-nets, which are called axial C-hedra. The obtained result can also be usedto construct their semi-discrete analogs. Moreover, we identify a novelsubclass within the determined class of (semi-)discrete axial cone-nets, whosemembers are named axial P-nets as they fulfill the proportion (P) of theintercept theorem. Known special cases of these axial P-nets are the smooth anddiscrete conic crease patterns with reflecting rule lines. By using aparallelism operation one can even generalize axial P-nets. The resultinggeneral P-nets constitute a rich novel class of continuous flexible(semi-)discrete surfaces, which allow direct access to their spatial shapes bythree control polylines. This intuitive method makes them suitable fortransformable design tasks using interactive tools.