Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From axial C-hedra to general P-nets

Georg Nawratil

TL;DR

This work provides a full classification of continuous flexible discrete axial cone-nets (axial C-hedra) and their semi-discrete analogs, and identifies a novel subclass, axial P-nets, defined by a proportionality condition between three key parameters. It reduces the problem to an overconstrained planar linkage built from three consecutive conical strips and derives necessary and sufficient conditions for a $1$-DoF mobility, framed by three projective maps that describe the motion. The authors present a constructive scheme to assemble continuous flexible $(\text{semi-})$discrete cone-nets and extend to semi-discrete surfaces, with axial P-nets arising via a parallelism operation and connected to conic crease patterns with reflecting rule lines. The results offer an intuitive, three-control-polyline design framework suitable for transformable design workflows and potential integration into interactive tools like Scutes, while outlining open questions on non-axial C-hedra and broader generalizations.

Abstract

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.

From axial C-hedra to general P-nets

TL;DR

This work provides a full classification of continuous flexible discrete axial cone-nets (axial C-hedra) and their semi-discrete analogs, and identifies a novel subclass, axial P-nets, defined by a proportionality condition between three key parameters. It reduces the problem to an overconstrained planar linkage built from three consecutive conical strips and derives necessary and sufficient conditions for a -DoF mobility, framed by three projective maps that describe the motion. The authors present a constructive scheme to assemble continuous flexible discrete cone-nets and extend to semi-discrete surfaces, with axial P-nets arising via a parallelism operation and connected to conic crease patterns with reflecting rule lines. The results offer an intuitive, three-control-polyline design framework suitable for transformable design workflows and potential integration into interactive tools like Scutes, while outlining open questions on non-axial C-hedra and broader generalizations.

Abstract

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.
Paper Structure (6 sections, 1 theorem, 8 equations, 2 figures)

This paper contains 6 sections, 1 theorem, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation, Outline and Review
  3. Determination of axial C-hedra
  4. The planar linkage $\mathcal{L}$
  5. Construction of continuous flexible (semi-)discrete cone-nets
  6. Conclusion and future work

Key Result

theorem 1

The linkage $\mathcal{L}$ has an overconstrained 1-DoF mobility if and only if each sublinkage with index $j$ for $j\in\left\{1,\ldots, p\right\}$ belongs to one of the following cases defined by the initial linkage $L_0$:

Figures (2)

  • Figure 1: Schematic sketch of three consecutive conical strips of an axial C-hedra in an axonometric view (left) and the corresponding overconstrained planar linkage $\mathcal{L}$ located in the $xz$-plane (right) where the sublinkages $L_0$, $L_1$ and $L_i$ are highlighted in orange, green and blue, respectively. In order to save space the pictures are rotated by 90 degrees.
  • Figure 2: Illustration of the central scaling $\sigma_1$ (left), perspective collineation $\sigma_2$ (center) and central perspectivity $\sigma_3$ (right), respectively. In order to save space the pictures are rotated by 90 degrees.

Theorems & Definitions (6)

  • remark 1
  • theorem 1
  • remark 2
  • remark 3
  • remark 4
  • remark 5