Construction and Fabrication of Reversible Shape Transforms

ACM Transactions on Graphics (Proceedings of SIGGRAPH ASIA)

Shuhua Li1,2         Ali Mahdavi-amiri2         Ruizhen Hu3*         Han Liu2,4         Chanqing Zou5         Oliver Van Kaick4         Xiuping Liu1         Hui Huang3*         Hao Zhang2

1Dalian University of Technology          2Simon Fraser University          3Shenzhen University          4Carleton University          5University of Maryland, College Park

Fig. 1. We introduce a fully automatic algorithm to construct reversible hinged dissections: the crocodile and the Crocs shoe can be inverted inside-out and transformed into each other, bearing slight boundary deformation. The complete solution shown was computed from the input (left) without user assistance. We physically realize the transform through 3D printing (right) so that the pieces can be played as an assembly puzzle.


We study a new and elegant instance of geometric dissection of 2D shapes: reversible hinged dissection, which corresponds to a dual transform between two shapes where one of them can be dissected in its interior and then inverted inside-out, with hinges on the shape boundary, to reproduce the other shape, and vice versa.We call such a transform reversible inside-out transform or RIOT. Since it is rare for two shapes to possess even a rough RIOT, let alone an exact one, we develop both a RIOT construction algorithm and a quick filtering mechanism to pick, from a shape collection, potential shape pairs that are likely to possess the transform. Our construction algorithm is fully automatic. It computes an approximate RIOT between two given input 2D shapes, whose boundaries can undergo slight deformations, while the filtering scheme picks good inputs for the construction. Furthermore, we add properly designed hinges and connectors to the shape pieces and fabricate them using a 3D printer so that they can be played as an assembly puzzle. With many interesting and fun RIOT pairs constructed from shapes found online, we demonstrate that our method significantly expands the range of shapes to be considered for RIOT, a seemingly impossible shape transform, and offers a practical way to construct and physically realize these transforms.

Fig. 5. Overview of our work on reversible hinged dissections. Given a shape collection, we compute reversibility scores to quickly assess how likely two shapes possess a reversible transform. (a) Scores of diferent shapes with respect to the bird. Given a promising pair of shapes, e.g., the bird and the hat in (b1), we construct an approximate reversible inside-out transform through several steps: candidate trunk selection (b2), trunk pair selection (b2), and slight boundary deformation (b3)-(b4) to perfect the transform. The shapes can finally be textured (b5) and fabricated.

Fig. 9. When deforming shape P, we fix its candidate trunk TP (a) and conjugate trunk TQ (enlarged) (b). The goal is to eliminate regions outside TQ (b to c), the overlaps (c to d), and gaps (d to e) inside TQ. The user is allowed to directly draw new segments (red segments in g) on the deformed shape (f); The deformed shape and dissection curves inside TQ are updated (h).

Fig. 14. A gallery of reversible shape transforms computed fully automatically by our algorithm. For each pair, we show the input shapes in silhouette images and the resulting, possibly deformed, shapes which induce a RIOT in texture. Hinged dissections are shown in a circular sequence.


We thank the anonymous reviewers for their valuable comments. This work was supported in parts by China Scholarship Council, NSERC Canada (611370, 611649, 2015-05407), NSFC (61528208, 61602311, 61522213, 61432003, 61370143), GD Science and Technology Program (2015A030312015), Shenzhen Innovation Program (JCYJ20170302153208613, KQJSCX20170727101233642), and gift funds from Adobe. We would also like to thank Richard Bartels, and Akshay Gadi Patil for proofreading and helpful comments and Kai Yang for his artistic works to texture our results.


title = {Construction and Fabrication of Reversible Shape Transforms},
author = {Shuhua Li and Ali Mahdavi-amiri and Ruizhen Hu and Han Liu and Chanqing Zou and Oliver Van Kaick and Xiuping Liu and Hui Huangand  Hao Zhang},
journal = {ACM Transactions on Graphics (Proc. SIGGRAPH ASIA)},
volume = {37},
number = {6},
pages = {190:1--190:14},  
year = {2018},