Unfolding via Progressive Mesh Approximation

TL;DR

The paper proposes a progressive mesh unfolding framework that couples edge-collapse-based simplification with Tabu Unfolding to search for overlap-free unfoldings. By allowing the approximating mesh to change during unfolding and then reversing the simplification, the method can overcome not-unfoldability and deliver faster, more reliable results than prior approaches. Empirical evaluations on a Thingi10k subset show competitive coverage and stable aspect ratios, with improved reliability and significant timing advantages, particularly using Quadric Error-based collapses. The approach eliminates the need for mesh segmentation and broadens practical applicability to manufacturing pipelines for paper-based folding, architecture, and robotics.

Abstract

When folding a 3D object from a 2D material like paper, typically only an approximation of the original surface geometry is needed. Such an approximation can effectively be created by a (progressive) mesh simplification approach, e.g. using an edge collapse technique. Moreover, when searching for an unfolding of the object, this approximation is assumed to be fixed. In this work, we take a different route and allow the approximation to change while searching for an unfolding. This way, we increase the chances to overcome possible ununfoldability issues. To join the two concepts of mesh approximation and unfolding, our work combines the edge collapsing mesh simplification technique with a Tabu Unfolder, a robust mesh unfolding approach. We empirically show that this strategy performs faster and that it is more reliable than prior state of the art methods.

Figures (13)

• Figure 1: Two different unfoldings of a cube.
• Figure 2: A folded triangulated cube and the corresponding unfolding. The unfold-tree is indicated in blue. The cut-tree over the edges of the mesh is visualized in dark red. In the unfolding, the cut-tree is the boundary and is not colored.
• Figure 3: A visualization of our pipeline applied to the stanford bunny with 2000 faces. In the first phase, edges are collapsed (top row, green arrows). Then, the low resolution mesh is initially unfolded (right column, blue arrow). In the third phase, edges are uncollapsed while the unfolding is kept overlap-free (bottom row backwards, orange arrows).
• Figure 4: Inserting triangles into the unfold-tree after uncollapsing an edge. The important parts of the unfold-tree are highlighted in purple and the uncollapsed edge is highlighted in red.
• Figure 5: The Utah Tea Pot with 800 faces.