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Research Papers

Edge-Compositions of 3D Surfaces

[+] Author and Article Information
Cynthia Sung

e-mail: crsung@mit.edu

Erik D. Demaine

e-mail: edemaine@mit.edu

Martin L. Demaine

e-mail: mdemaine@mit.edu

Daniela Rus

e-mail: rus@mit.edu

Computer Science and Artificial
Intelligence Laboratory,
Massachusetts Institute of Technology,
Cambridge, MA 02139

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 1, 2013; final manuscript received May 29, 2013; published online September 24, 2013. Assoc. Editor: Larry L. Howell.

J. Mech. Des 135(11), 111001 (Sep 24, 2013) (9 pages) Paper No: MD-13-1048; doi: 10.1115/1.4025378 History: Received February 01, 2013; Revised May 29, 2013

Origami-based design methods enable complex devices to be fabricated quickly in plane and then folded into their final 3D shapes. So far, these folded structures have been designed manually. This paper presents a geometric approach to automatic composition of folded surfaces, which will allow existing designs to be combined and complex functionality to be produced with minimal human input. We show that given two surfaces in 3D and their 2D unfoldings, a surface consisting of the two originals joined along an arbitrary edge can always be achieved by connecting the two original unfoldings with some additional linking material, and we provide a polynomial-time algorithm to generate this composite unfolding. The algorithm is verified using various surfaces, as well as a walking and gripping robot design.

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Figures

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Fig. 2

Q can be unfolded into P if Q is the image of a folded state φF(P)

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Fig. 3

A fold f has a fold line and a fold angle

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Fig. 1

Left: Walking and gripper robots folded out of the patterns shown. Right: The composition, a walking-gripping robot, whose unfolding was designed manually. Our goal is to generate such an unfolding automatically. (Credit: Robots were designed by Cagdas Onal and Michael Tolley.)

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Fig. 4

Two convex polygons placed next to each other are guaranteed not to intersect

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Fig. 5

Algorithm 1: Constructing pleats to follow a path. (a) The edge eP and the path p to follow. (b) Reflected edges ei (solid) and perpendicular bisectors ei⊥ (dotted). (c) Resulting pleats. Types 1, 2, and 3 self-intersections are shaded. (d) Pleats with self-intersections corrected. (e) Folded state of pleats. All ei coincide.

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Fig. 6

Algorithm 2: Bridging an edge on the boundary of the unfolding to the boundary of the convex hull. (a) Original fold pattern. The edge to join eP is bold and the convex hull is shown in gray. (b) The region PkCH (shaded) and its straight skeleton. (c) The path p from eP to the boundary of the convex hull. (d) Pleats tiled along the path. (e) Output unfolding with the bridge added.

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Fig. 7

(a) A bridge (shaded) that intersects with (P, F). (b) The offending region is removed.

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Fig. 8

A long pleat. Top: The unfolding with offending area (shaded). Bottom: Folded state. (a) In the folded state, the pleat protrudes outside the adjacent face. (b) It can be crimped to not interfere with other folds and (c) trimmed to avoid protrusions.

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Fig. 9

Algorithm 3: Bridging an edge on the interior of the unfolding to the boundary of the convex hull. (a) Original fold pattern. The edge to join eP is bold and the convex hull is shown in gray. (b) The edge-adjacency graph with the path from eP to eb in bold. (c) Pleats attached to eb using Algorithm 2. (d) The accordion path and interior faces reflected over the boundary of the convex hull. The convex hull is also updated. (e) Pleats attached to erefP using Algorithm 2. (f) Output unfolding with the bridge added.

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Fig. 10

Example edge-adjacency graph

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Fig. 11

Fold patterns generated by this algorithm. Top: Input fold patterns for the polyhedral complexes to join. The bold edges are the edges to join. Second row: The generated composite unfolding. Bridges constructed by our algorithm are shaded. Third row: The folded state of the composite unfolding. Fourth row: Physical model of the composition with just the bridge folded. Bottom: Physical models of the input surfaces and the composition folded from poster board.

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