Mesh Simplification For Unfolding

TL;DR

The paper tackles the intractable problem of finding a single-patch, overlap-free isometric unfolding for arbitrary 3D meshes by proposing a geometric relaxation: minimally modify the input mesh to admit an unfoldable single patch. It introduces an unfolding-aware pipeline that alternates vertex-level geometry adjustments with unfolding-preserving edge collapses, followed by a post-processing step that couples the original and unfolded meshes to minimize distortion while removing overlaps. Key contributions include unfolding-aware vertex manipulation, unfolding-aware decimation that preserves the unfolding's spanning tree, and a practical post-processing framework, all validated on large datasets and demonstrated via paper-based fabrications. Results show substantial improvements over prior methods in success rates and mesh fidelity, enabling practical fabrication workflows and suggesting directions for integrating user constraints and extending to multiple patches.

Abstract

We present a computational approach for unfolding 3D shapes isometrically into the plane as a single patch without overlapping triangles. This is a hard, sometimes impossible, problem, which existing methods are forced to soften by allowing for map distortions or multiple patches. Instead, we propose a geometric relaxation of the problem: we modify the input shape until it admits an overlap-free unfolding. We achieve this by locally displacing vertices and collapsing edges, guided by the unfolding process. We validate our algorithm quantitatively and qualitatively on a large dataset of complex shapes and show its proficiency by fabricating real shapes from paper.

Figures (14)

• Figure 1: Existing methods fail at consistently producing simple, non-overlapping unfoldings of complex shapes (middle). Instead, our method finds a close approximation to the input shape that admits a simple unfolding (right). In practice, this massively increases the number of shapes that can be fabricated from a single sheet of material (see quantitative bar chart on the left).
• Figure 2: Not only are simple unfoldings hard to compute; in some cases, they can be theoretically shown not to exist demaine2020acutely. This motivates us to find a close approximation to the input shape that admits an unfolding.
• Figure 3: The potential single-patch unfoldings of a given mesh can be represented by the spanning trees of its dual graph. The choice of spanning tree (here, random) can greatly affect the shape of the unfolding and the number of overlaps in it.
• Figure 4: A convex vertex unfolds in 2D without overlaps with one cut, but a saddle vertex's single cut always causes overlapping triangles.
• Figure 5: Our algorithm is composed of several steps, all of which are crucial to obtaining a single patch unfolding that is non-overlapping (see first two subfigures) and differs as little as possible from the original mesh (see regions circled in brown).