Interactive Hexagonal Mesh Editing for Beadwork Design
Yuki Igarashi∗
University of Tsukuba

1

Takeo Igarashi
The University of Tokyo / JST ERATO

Introduction

Beadwork is the art of connecting beads together with wire.
Igarashi et al. [2012] presented an interactive beadwork design system called Beady to help non-professionals design their own 3D
beadwork. They observed that existing beadwork designs, especially large ones, typically consist of hexagonal faces. This is probably because a hexagonal mesh (honeycomb lattice) is the most efficient structure for holding flat surfaces with minimal support materials. After conducting physical simulations, they also found thata near-hexagonal mesh, obtained as the dual of a triangular mesh,yields a more aesthetically pleasing beadwork model. However,
the interactive modeling interface of the original Beady system did
not consider this. Thus the user had to carefully combine various
editing operations to construct a near-hexagonal polyhedron. Existing 3D modeling software is also inconvenient for near-hexagonalmesh modeling. Therefore, we introduce mesh-editing operations
specifically designed for creating near-hexagonal polyhedra. Bycombining the original Beady interface with our method, the user
can more easily design near-hexagonal polyhedra.

2Jun Mitani
University of Tsukuba / JST ERATO

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(b)

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Figure 1: Merge faces. The system takes the dual (b), collapses theedge (c), and again takes the dual (d).

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Figure 2: Flip faces. The system takes the dual (b), flips the edge(c), and again takes the dual (d).

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Figure 3: Insert faces. The system opens the mesh (b) and inserts
new edges (c).

Hexagonal Editing Operations

The user designs a near-hexagonal mesh via a gestural user interface based on the original Beady interface. We provide four special mesh-editing operations (merge faces, flip faces, insert faces
and delete faces) for creating near-hexagonal meshes. Because it issometimes easier to use standard editing operations to make smallchanges, the hexagonal mesh-editing operations are included as anoption. The user carries out these operations while holding down
the Shift key.
Merge faces: Drawing a line from one face to another face with ashared edge between them merges these faces (Fig. 1). The systemtakes the dual of the mesh and collapses the edge shared by thetarget faces. Then the system again takes the dual of the mesh.
Flip faces: When the user draws a line from one face to anotherface, and the two faces are bridged by an edge, the system flips thefaces (Fig. 2). The system takes the dual of the mesh and flips thebridging edge. Then the system again takes the dual of the mesh.
Insert faces: Pressing the mouse button with the cursor on an edge
and moving the cursor with the mouse button held down selects theedges touched by the cursor. This operation inserts new faces along
the selected edges (Fig. 3). The system opens the mesh along theselected edges and inserts new faces in the gap, in such a way thatthe valences of the vertices along the gap are as close to three aspossible. We formulate this as an optimization problem that finds
a best match among the vertices on the two sides of the gap. The
system inserts new edges connecting the matched pairs. The costfunction is given by Σv∈v0 |valence(v) − 3|, where v0 denotes thevertices along the gap, and valence(v) denotes the valence of vertex v after inserting the edges. We impose the constraint that thematching order is monotonic, and solve the optimization problem
using a beam search.
Delete faces: Pressing the mouse button with the cursor on a faceand moving the cursor with the mouse button held down selects the∗ e-mail: yukim@acm.org

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(b)

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Figure 4: Delete faces. The system deletes faces (b) and merges
the matched vertex pairs (c).

faces touched by the cursor (Fig. 4). The system deletes the selected faces and closes the gap by merging the vertices along thegap. This is again formulated as an optimization problem that finds
a best match among the vertices on the two sides of the gap. The
system merges the matched pairs. The cost function is given byΣv∈v1 |valence(v) − 3|, where v1 denotes the vertices after themerge, and valence(v) denotes the valence of the merged vertex
v. We again impose the constraint that the matching order is monotonic, and solve the optimization problem using a beam search.

3 ResultsFigure 5 shows modeling sequences using the near-hexagonal
mesh-editing operations. The user can design these near-hexagonalmodels without expending any additional effort to make themhexagonal.

Figure 5: Example of modeling sequences using the nearhexagonal mesh-editing operations.

References
I GARASHI , Y., I GARASHI , T., AND M ITANI , J. 2012. Beady: Interactive beadwork design and construction. ACM Trans. Graph.
31, 4.