Making Papercraft Toys from Meshes using Strip-based Approximate Unfolding

1. Making Papercraft Toys from Meshesusing Strip-based Approximate Unfolding Jun Mitani* Hiromasa Suzuki† University of Tokyo Siggraph2004

2. Outline • Introduction • Triangle strip • Detailed Method • Results • Conclusion and Future Work

3. Introduction • Motivation • Related work • Elber 1995; Pottmann&Farin 1995; Hoschek 1998 • These methods approximate parametric surfaces such as B-spline surfaces or rational Bézier surfaces by sets of ruled surfaces such as parts of cones or cylinders. • Chen et al. 1999 They are hard to handle free form models of triangulated meshes Elber 1995 Hoschek 1998

4. Introduction • Sorkine et al.2002;Lévy et al. 2002;Sheffer 2002 • Mesh parameterization • Generate the pattern by placing triangular faces on a plane Sheffer 2002 Lévy et al. 2002 Sorkine et al. 2002

5. Introduction

6. Triangle strip • Unfolding a polyhedron requires cuts, butwe can easily unfold a triangle strip without cuts. • Triangle strips canbe connected by branching triangles to form a triangle tree. • Ourstripsaregenerated by changing the geometry and topology of the originalmeshes

7. Detailed Method

8. Detailed Method • Feature Line Extraction and Partitioning • Generation of Zonal Regions • Addition of Internal Cut-Lines • Smoothing of Cutting Lines • Simplification to Generate Strips • Unfolding and Packing

9. Detailed Method

10. Detailed Method • Feature Line Extraction and Partitioning • Generation of Zonal Regions • Addition of Internal Cut-Lines • Smoothing of Cutting Lines • Simplification to Generate Strips • Unfolding and Packing

11. Feature Line Extraction and Partitioning • Garland et al. 2001; Katz and Tal 2003; Lévy et al. 2002 • Extracts feature lines • Lines withsharp edges and which are longer than some predefined length.

12. Feature Line Extraction and Partitioning • Merge small charts algorithm 1. Select the chart C with the smallest number of triangles. If Chas more than predefined number of triangles (e.g. 3% of alltriangles), finish. 2. For each chart H other than C, count the number of edgeslying on the border between H and C (excluding theextracted feature lines). 3. Merge C with the chart for which the count is maximal, andrepeat from step 1.

13. Detailed Method • Feature Line Extraction and Partitioning • Generation of Zonal Regions • Addition of Internal Cut-Lines • Smoothing of Cutting Lines • Simplification to Generate Strips • Unfolding and Packing

14. Generation of Zonal Regions • Assign a value toeach triangle • Thevalue is the topological distance from the nearest partborder or feature line to thetriangle. • Zonal region borders are added along edges that connect triangles with assigned values nw and nw+1 (for n=1,2,3,…, and w is a positive integer)

15. Generation of Zonal Regions • Segment the part by placing region borders onedges • Mergenarrowinternal areas • The borders of zonal areas：border cut-lines

16. Detailed Method • Feature Line Extraction and Partitioning • Generation of Zonal Regions • Addition of Internal Cut-Lines • Smoothing of Cutting Lines • Simplification to Generate Strips • Unfolding and Packing

17. Addition of Internal Cut-Lines feature cut-line

18. Addition of Internal Cut-Lines

19. Addition of Internal Cut-Lines • Algorithm to extract core lines (有時間在探討) 1. Add all triangles in the region to a list T, and make an outerloop L of the region that is a list of edges incounterclockwise order. 2. Update L by removing a triangle from T. (1) (2) (3) (4) (5) (6)

20. Addition of Internal Cut-Lines 3. Repeat step 2 until Tis empty. • These lines are too complicated to craft , so we simplify each core line. (1) (2)

21. Addition of Internal Cut-Lines • Algorithm to simplify core lines 1.Create an edge-vertex tree of the core lines. (a) Make a list L of the vertices that are leaves ofthe edge-vertextree. (b) Repeatedly remove one vertex in L . When the number of vertices is reduced to a predefined ratio (e.g. 15%), go tostep 2. If L becomes empty, update the edge-vertex treeand repeat from (a).

22. Addition of Internal Cut-Lines 2. If there are any vertices nearer than a predefined distance(e.g. 0.3w) to the outer loop, remove the edges that connectthese vertices from core line. • The resulting core lines have been simplified. • We call these corelines center cut-lines. (2) (3)

23. Detailed Method • Feature Line Extraction and Partitioning • Generation of Zonal Regions • Addition of Internal Cut-Lines • Smoothing of Cutting Lines • Simplification to Generate Strips • Unfolding and Packing

24. Smoothing of Cutting Lines • Connectivity smoothing • Geometrical smoothing

25. Detailed Method • Feature Line Extraction and Partitioning • Generation of Zonal Regions • Addition of Internal Cut-Lines • Smoothing of Cutting Lines • Simplification to Generate Strips • Unfolding and Packing

26. Simplification to Generate Strips • Cohen [1999]

27. Detailed Method • Feature Line Extraction and Partitioning • Generation of Zonal Regions • Addition of Internal Cut-Lines • Smoothing of Cutting Lines • Simplification to Generate Strips • Unfolding and Packing

28. Unfolding and Packing • Place one of triangles in thestrip on plane • Recursively add those triangles connected totriangles already in the plane • When they intersect, wedivide the unfolded strip into two. • The problem of packing pieces has been studied elsewhere (see [Milenkovic 1999])

29. Results (1) (2) (3) (4) (5) (9) (6) (7) (8)

30. Results

31. Conclusion and Future Work • Effectiveness of the keep the smoothness of original meshes. • Cannot specify the approximation tolerance to the input mesh model

32. 補充-Detect Feature • Compute a sharpness criterion on the edges. • Choose a threshold so that a certain proportion of the edges is filtered out. • Grow a feature curve.

33. 補充-Chart圖示 紅線：feature line 黑線：輪廓圖的邊 橘線：要計算的邊數