Area preserving Combescure transformations

TL;DR

The paper classifies all deformable quad nets in isotropic 3-space that admit a continuous family of area-preserving Combescure transformations. It shows that deformable nets fall into two exclusive classes: (i) cone-cylinder nets formed by two interleaved cone-cylinder structures, and (ii) Koenigs nets with Christoffel duals carrying equal face areas; these findings extend from 2×2 to general m×n nets and persist in smooth analogs. Through metric duality, the discrete isotropic results translate into flexible nets in isotropic geometry, offering explicit constructions and a pathway to Euclidean realizations via optimization. The work highlights two contrasting behaviors: class (i) yields visually smooth cone-cylinder structures, while class (ii) tends to crumple, informing how isotropic models can seed design and numerical strategies for Euclidean flexible nets.

Abstract

Motivated by the design of flexible nets, we classify all nets of arbitrary size m x n that admit a continuous family of area-preserving Combescure transformations. There are just two different classes. The nets in the first class are special cases of cone nets that have been recently studied by Kilian, Mueller, and Tervooren. The second class consists of Koenigs nets having a Christoffel dual with the same areas of corresponding faces. We apply isotropic metric duality to get a new class of flexible nets in isotropic geometry. We also study the smooth analogs of the introduced classes.

Figures (18)

• Figure 1: A sequence of deformations of a Q-net from class (i) in $\mathbb R^3$. Corresponding edges are parallel and corresponding faces have equal areas. For a net from class (i), the two faces of each $1\times2$ sub-net or each $2\times1$ sub-net are affine symmetric. See Theorem \ref{['th-mxn']}
• Figure 1: A $2\times 2$ net (solid lines) and its deformation (dashed lines). Corresponding edges are parallel and corresponding faces (colored with the same color) have equal areas. See Definition \ref{['def-deformable-net']}.
• Figure 1: Left: An L-shaped net of size $6\times6$. Right: The unique deformable $6\times6$ net containing the L-shaped net. See Corollary \ref{['cor-L']}.
• Figure 1: A cone-cylinder net (left) and its metric dual (right). The latter is a flexible surface in isotropic geometry having planar discrete parameter lines, one family lying in isotropic planes.
• Figure 2: A sequence of deformations of a net from class (ii) in $\mathbb R^3$. Any two neighboring faces have equal so-called opposite ratios with respect to the common edge. See Section \ref{['ssec-statement-deformable-2x2']} and Theorem \ref{['th-mxn']}

Theorems & Definitions (65)

• Definition 2.1
• Theorem 2.2
• Example 2.3
• Proof 1
• Proposition 2.4
• Proof 2: Proof
• Proposition 2.5
• Proposition 2.6
• Proof 3
• Proposition 2.7