1. Joining Unfoldings of 3-D Surfaces
IDETC/CIE 2013
Joining Unfoldings of 3-D Surfaces
Cynthia Sung, Erik D. Demaine, Martin L. Demaine, Daniela Rus
September 23, 2013
15 min

2. Related Work in Unfolding
Computational Geometry
Polyhedron unfolding (Schlichenrieder 1997, Demaine et al. 2002, Bern 2003, Lucier 2006)
Unfolding by composition (Mitani 2011, Cheng & Cheong 2012)
Sheet Metal Design
Automated design (Wang 1997, Bush & Sèquin 1999, Patel & Campbell 2010)
DETC Joining Unfoldings of 3-D Surfaces

3. Our Contribution
An algorithm that computes a one-piece non-self-intersecting edge-composition of two folded surfaces.
DETC Joining Unfoldings of 3-D Surfaces

4. Edge-Composition Problem
Given: Two unfoldings and an edge on each +
DETC Joining Unfoldings of 3-D Surfaces

5. Edge-Composition Problem
Given: Two unfoldings and an edge on each
Find: An unfolding of the combined surface, where the two originals meet at the given edges
More technical description in paper + 
DETC Joining Unfoldings of 3-D Surfaces

6. Main Insight
If the edges to join are on the boundaries of the convex hulls of the unfoldings, then place the edges coincident.
DETC Joining Unfoldings of 3-D Surfaces

7. Main Insight
If the edges to join are on the boundaries of the convex hulls of the unfoldings, then place the edges coincident.
DETC Joining Unfoldings of 3-D Surfaces

8. Main Insight
If the edges to join are on the boundaries of the convex hulls of the unfoldings, then place the edges coincident.
DETC Joining Unfoldings of 3-D Surfaces

9. Main Insight
If the edges to join are not on the boundaries of the convex hulls of the unfoldings, then construct a bridge.
Add real unfoldings here
DETC Joining Unfoldings of 3-D Surfaces

10. Main Insight
If the edges to join are not on the boundaries of the convex hulls of the unfoldings, then construct a bridge.
Add real unfoldings here
DETC Joining Unfoldings of 3-D Surfaces

11. Main Insight
If the edges to join are not on the boundaries of the convex hulls of the unfoldings, then construct a bridge.
Add real unfoldings here
DETC Joining Unfoldings of 3-D Surfaces

12. Main Insight
If the edges to join are not on the boundaries of the convex hulls of the unfoldings, then construct a bridge.
Add real unfoldings here
DETC Joining Unfoldings of 3-D Surfaces

13. Main Insight
If the edges to join are not on the boundaries of the convex hulls of the unfoldings, then construct a bridge.
Add real unfoldings here
DETC Joining Unfoldings of 3-D Surfaces

14. Main Insight
If the edges to join are not on the boundaries of the convex hulls of the unfoldings, then construct a bridge.
Add real unfoldings here
DETC Joining Unfoldings of 3-D Surfaces

15. Main Insight
If the edges to join are not on the boundaries of the convex hulls of the unfoldings, then construct a bridge.
Add real unfoldings here
DETC Joining Unfoldings of 3-D Surfaces

16. Main Insight
If the edges to join are not on the boundaries of the convex hulls of the unfoldings, then construct a bridge.
Add real unfoldings here
DETC Joining Unfoldings of 3-D Surfaces

17. Constructing a Bridge Case 1: Edges on the Boundary
Input: unfolding, edge
edge to join
convex hull boundary
DETC Joining Unfoldings of 3-D Surfaces

18. Constructing a Bridge Case 1: Edges on the Boundary
Input: unfolding, edge Compute medial axis
medial axis
DETC Joining Unfoldings of 3-D Surfaces

19. Constructing a Bridge Case 1: Edges on the Boundary
Input: unfolding, edge Compute medial axis Find path to the boundary of the convex hull
path
DETC Joining Unfoldings of 3-D Surfaces

20. Constructing a Bridge Case 1: Edges on the Boundary
Input: unfolding, edge Compute medial axis Find path to the boundary of the convex hull Overlay pleats Remove intersections
pleats
DETC Joining Unfoldings of 3-D Surfaces

21. Constructing a Bridge Case 1: Edges on the Boundary
Input: unfolding, edge Compute medial axis Find path to the boundary of the convex hull Overlay pleats Remove intersections Output: bridged unfolding
DETC Joining Unfoldings of 3-D Surfaces

22. Pleat Construction
DETC Joining Unfoldings of 3-D Surfaces

23. Constructing a Bridge Case 2: Edges on the Interior
Input: unfolding, edge
edge to join
convex hull boundary
DETC Joining Unfoldings of 3-D Surfaces

24. Constructing a Bridge Case 2: Edges on the Interior
Input: unfolding, edge Find path to the boundary
path
DETC Joining Unfoldings of 3-D Surfaces

25. Constructing a Bridge Case 2: Edges on the Interior
Input: unfolding, edge Find path to the boundary Bridge to the boundary of the convex hull
initial bridge
DETC Joining Unfoldings of 3-D Surfaces

26. Constructing a Bridge Case 2: Edges on the Interior
Input: unfolding, edge Find path to the boundary Bridge to the boundary of the convex hull Reflect traversed faces
reflected faces
DETC Joining Unfoldings of 3-D Surfaces

27. Constructing a Bridge Case 2: Edges on the Interior
Input: unfolding, edge Find path to the boundary Bridge to the boundary of the convex hull Reflect traversed faces
added bridge
DETC Joining Unfoldings of 3-D Surfaces

28. Constructing a Bridge Case 2: Edges on the Interior
Input: unfolding, edge Find path to the boundary Bridge to the boundary of the convex hull Reflect traversed faces Output: bridged unfolding
DETC Joining Unfoldings of 3-D Surfaces

29. Overlaps with the Unfolding
DETC Joining Unfoldings of 3-D Surfaces

30.  Example Compositions
DETC Joining Unfoldings of 3-D Surfaces

31.  Example Compositions
DETC Joining Unfoldings of 3-D Surfaces

32. Current Work
DETC Joining Unfoldings of 3-D Surfaces