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Low-distortion planar embedding of rod-based structures

Mark Yan Lok Yip, Gary P. T. Choi

TL;DR

The paper addresses the challenge of representing 3D rod-based structures in a low-distortion plane while preserving geometry critical for design and fabrication. It introduces a two-stage pipeline: (1) compute an initial planar embedding $f_0$ via a Tutte-style solution of $Loldsymbol{v}=0$, and (2) solve a constrained planar shape optimization $g$ that enforces length-preserving constraints $E_{ ext{L}}(e_i)= rac{L_i}{l_i}-1=0$, angle-preserving constraints $E_{ ext{A}}(j)= ext{cos}( heta_{2D_j})- ext{cos}( heta_{3D_j})=0$, and a no-overlap constraint $E_{ ext{O}}= extstyleig( ext{Area}(T_i)ig) - ext{Area}(oldsymbol{eta})=0$, with explicit gradient formulas to enable efficient, robust optimization. The approach is extended to hybrid rod-and-surface structures and validated through 2D-to-3D morphing using a spring-energy deployment model, demonstrating accurate preservation of local geometry and bijectivity (zero overlaps) across diverse shapes, including cloth-like and Sophie-surface examples. This framework supports practical manufacturing and design workflows by providing compact, bijective planar representations that can be morphed back to the target 3D forms. The work also outlines an overlap-correction scheme to relax strict no-overlap enforcement when necessary, ensuring convergence and applicability to complex geometries. Overall, the method offers a principled, geometry-preserving flattening of rod-based assemblies with direct applicability to fabrication and deployment tasks.

Abstract

Rod-based structures are commonly used in practical applications in science and engineering. However, in many design, analysis, and manufacturing tasks, handling the rod-based structures in three dimensions directly is generally challenging. To simplify the tasks, it is usually more desirable to achieve a two-dimensional representation of the rod-based structures via some suitable geometric mappings. In this work, we develop a novel method for computing a low-distortion planar embedding of rod-based structures. Specifically, we identify geometrical constraints that aim to preserve key length and angle quantities of the 3D rod-based structures and prevent the occurrence of overlapping rods in the planar embedding. Experimental results with a variety of rod-based structures are presented to demonstrate the effectiveness of our approach. Moreover, our method can be naturally extended to the design and mapping of hybrid structures consisting of both rods and surface elements. Altogether, our approach paves a new way for the efficient design and fabrication of novel three-dimensional geometric structures for practical applications.

Low-distortion planar embedding of rod-based structures

TL;DR

The paper addresses the challenge of representing 3D rod-based structures in a low-distortion plane while preserving geometry critical for design and fabrication. It introduces a two-stage pipeline: (1) compute an initial planar embedding via a Tutte-style solution of , and (2) solve a constrained planar shape optimization that enforces length-preserving constraints , angle-preserving constraints , and a no-overlap constraint , with explicit gradient formulas to enable efficient, robust optimization. The approach is extended to hybrid rod-and-surface structures and validated through 2D-to-3D morphing using a spring-energy deployment model, demonstrating accurate preservation of local geometry and bijectivity (zero overlaps) across diverse shapes, including cloth-like and Sophie-surface examples. This framework supports practical manufacturing and design workflows by providing compact, bijective planar representations that can be morphed back to the target 3D forms. The work also outlines an overlap-correction scheme to relax strict no-overlap enforcement when necessary, ensuring convergence and applicability to complex geometries. Overall, the method offers a principled, geometry-preserving flattening of rod-based assemblies with direct applicability to fabrication and deployment tasks.

Abstract

Rod-based structures are commonly used in practical applications in science and engineering. However, in many design, analysis, and manufacturing tasks, handling the rod-based structures in three dimensions directly is generally challenging. To simplify the tasks, it is usually more desirable to achieve a two-dimensional representation of the rod-based structures via some suitable geometric mappings. In this work, we develop a novel method for computing a low-distortion planar embedding of rod-based structures. Specifically, we identify geometrical constraints that aim to preserve key length and angle quantities of the 3D rod-based structures and prevent the occurrence of overlapping rods in the planar embedding. Experimental results with a variety of rod-based structures are presented to demonstrate the effectiveness of our approach. Moreover, our method can be naturally extended to the design and mapping of hybrid structures consisting of both rods and surface elements. Altogether, our approach paves a new way for the efficient design and fabrication of novel three-dimensional geometric structures for practical applications.
Paper Structure (14 sections, 1 theorem, 44 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 14 sections, 1 theorem, 44 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Proposed method
  3. Initial embedding
  4. Shape optimization
  5. Length-preserving constraints
  6. Angle-preserving constraints
  7. No-overlap constraints
  8. The objective function and the constrained optimization formulation
  9. Overlap correction
  10. Summary
  11. Experimental results
  12. Extension to hybrid structures
  13. 2D-to-3D morphing process of the planar embeddings
  14. Conclusion

Key Result

Theorem 1

If the area enclosed by all boundary vertices in the shape optimization result is equal to the area sum of all triangles, then there is no overlap in the shape optimization result.

Figures (7)

  • Figure 1: An illustration of the proposed low-distortion planar embedding method for 3D rod-based structures in our work. Given a 3D rod-based structure with a prescribed target shape (left), our proposed method aims to produce a planar configuration of the structure that possesses low geometric distortion (right) with key rod length and angle quantities preserved, thereby facilitating the modelling and manufacturing of the rod-based structures for practical applications.
  • Figure 2: An illustration of the construction of triangulations for the no-overlap constraint. (a) An initial embedding of a rod-based structure and a Delaunay triangulation constructed on the set of all nodes. (b) An example of a deformed configuration of the planar embedding, with the associated triangulation induced from (a). It can be observed that the deformed rod-based structure does not contain overlaps, while several mesh overlaps can be found in the induced deformed triangulation, suggesting that the no-overlap constraint in Eq. \ref{['eqt:constraint_overlap']} is sufficient but not necessary for ensuring the no-overlap condition.
  • Figure 3: Quantification of the geometric distortion of the example shown in Fig. \ref{['fig:illustration']}. (a) The planar embedding result, with every rod segment color-coded with the length error. (b) The angle error at each intersection point of the rods.
  • Figure 4: Three examples of rod-based structures with different geometries (top) and the corresponding planar embeddings (bottom). (a) A doubly curved rod-based structure. (b) A rod-based structure with a more prominent height variation. (c) A rod-based structure with multiple peaks.
  • Figure 5: Two examples of rod-based structures representing different surfaces and the corresponding planar embedding. (a) The Cloth surface model. (b) The Sophie surface model.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1