Construction and deformation of P-hedra using control polylines

TL;DR

The paper addresses transforming and deforming quad-based PQ-surfaces by introducing P-hedra, which are reconstructible from three control polylines. It provides a practical reconstruction algorithm that builds an axial P-hedron via parallelism and $\sigma^\pm$ mappings, then recovers the full P-hedron from inverse translations. An explicit parametric framework for isometric deformations is developed, with a single motion parameter guiding both the planar linkage motion and the trajectory polyline, including clear flexion and bifurcation behavior. The work also explores developable patterns and P-hedral tubes as design primitives for reconfigurable architectures and metamaterial-inspired structures.

Abstract

In the 19th International Symposium on Advances in Robot Kinematics the author introduced a novel class of continuous flexible discrete surfaces and mentioned that these so-called P-hedra (or P-nets) allow direct access to their spatial shapes by three control polylines. In this follow-up paper we study this intuitive method, which makes these flexible planar quad surfaces suitable for transformable design tasks by means of interactive tools. The construction of P-hedra from the control polylines can also be used for an efficient algorithmic computation of their isometric deformations. In addition we discuss flexion limits, bifurcation configurations, developable/flat-foldable pattern and tubular P-hedra.

Figures (4)

• Figure 1: (left) Input of a general P-hedron: trajectory polyline $V_{0,0}, V_{1,0}, \ldots ,V_{m,0}$, direction polyline $D_0, D_1, \ldots , D_m$ and apex polyline $S_0, S_1^\pm ,\ldots ,S_{n-1}^\pm, S_{n}$. (center) Input of the associated axial P-hedron: trajectory polyline $V^{\circ}_{0,0}, V^{\circ}_{1,0}, \ldots ,V^{\circ}_{m,0}$, direction polyline $D^{\circ}_0, D^{\circ}_1, \ldots , D^{\circ}_m$ and apex polyline $S_0, S_1^\pm ,\ldots ,S_{n-1}^\pm, S_{n}$. (right) Illustration of the two constructions related to the sign of $S_{j+1}^\pm$.
• Figure 2: (left) Iterative composition of the two possible linear constructions $\sigma^+$ and $\sigma^-$ in $\pi^{\circ}_i$ with $V^{\circ}_{i,0}$ as starting point. Interpretation of the resulting point set as axial P-hedron (center) and the corresponding general P-hedron (right).
• Figure 3: (left) The point sets located in the plane $\pi^{\circ}_i$ for $i=0,\ldots m$ can be interpreted as planar linkage $L_i$ with mobility 1. If we rotate all planes $\pi^{\circ}_i$ for $i=1,\ldots m$ into $\pi^{\circ}_0$, we get the illustrated overconstrained planar linkage $\mathcal{L}$ discussed in nawratil. (middle-left) Bifurcation configuration of Remark \ref{['rem:no_bifur']}. Illustration of the linkage $L_0$ enclosing a parallelogram (middle-right) and an anti-parallelogram (right), respectively.
• Figure 4: Sequence of isometric deformation of the planar linkage $\mathcal{L}$ (top), axial P-hedron (middle) and general P-hedron (bottom). The corresponding animations can be downloaded from https://www.geometrie.tuwien.ac.at/nawratil/publications.html. The second column corresponds to the given configuration (parameter $t_*$) already illustrated in Figs. \ref{['fig2']} and \ref{['fig3']}-left. The third column illustrates a flexion limit and the fourth column displays the other branch for the parameter $t_*$. Therefore the planar linkage $\mathcal{L}$ has the same configuration in the second and fourth column. Note that both P-hedra of the fourth column have self-intersections.

Theorems & Definitions (4)

• remark 1
• remark 2
• remark 3
• remark 4