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

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) DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Our Contribution An algorithm that computes a one-piece non-self-intersecting edge-composition of two folded surfaces. DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Edge-Composition Problem Given: Two unfoldings and an edge on each + DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 +  DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Main Insight If the edges to join are on the boundaries of the convex hulls of the unfoldings, then place the edges coincident. DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Main Insight If the edges to join are on the boundaries of the convex hulls of the unfoldings, then place the edges coincident. DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Main Insight If the edges to join are on the boundaries of the convex hulls of the unfoldings, then place the edges coincident. DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Constructing a Bridge Case 1: Edges on the Boundary Input: unfolding, edge edge to join convex hull boundary DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Constructing a Bridge Case 1: Edges on the Boundary Input: unfolding, edge Compute medial axis medial axis DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Pleat Construction DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Constructing a Bridge Case 2: Edges on the Interior Input: unfolding, edge edge to join convex hull boundary DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Constructing a Bridge Case 2: Edges on the Interior Input: unfolding, edge Find path to the boundary path DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

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 DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Overlaps with the Unfolding DETC2013-12692 Joining Unfoldings of 3-D Surfaces

 Example Compositions DETC2013-12692 Joining Unfoldings of 3-D Surfaces

 Example Compositions DETC2013-12692 Joining Unfoldings of 3-D Surfaces

Current Work DETC2013-12692 Joining Unfoldings of 3-D Surfaces